UC-NRLF 


531 


PRACTICAL    TREATISE 


ON   THE 


PROPERTIES  OF 

CONTINUOUS  BRIDGES. 


BY  CHARLES  BENDER,  C.  E. 

MEMBER  OF  THE  AMERICAN  SOC.   CIVIL  ENGINEERS. 


NEW  YORK  : 

D.   VAN    NOSTRAND,    PUBLISHER, 

23  MURRAY  AND  27  WARREN  STREETS. 

1876 


COPYRIGHT, 

1876, 
BY     D.    VAN     NOSTRAND. 


CONTENTS. 


INTRODUCTION  AND  HISTORICAL  NOTES  ON  THE  THE- 
ORY OF  THE  ELASTIC  LINE,  AND  ESPECIALLY  OF 
CONTINUOUS  GIRDERS. 

I.— The  general  principle  of  continuity  of  girders  and 
trusses.  The  theory  reduced  to  a  combination  of  single 
spans,  the  theorem  of  three  moments  being  nothing  but 
its  algebraic  expression.  The  principle  of  continuity 
only  modifies  the  law  that  the  total  load  of  and  upon  a 
span  must  be  taken  up  by  its  |two  piers.  The  moments 
over  the  middle  piers  expressed  by  these  elastic  reac- 
tions. 

Correct  mode  of  calculation  of  deflections  of  trusses 
indicated. 

II.— The  modulus  of  elasticity  depending  upon  special  ex- 
periments. Tensile  experiments  by  Bornet,  Ardant, 
Hodgkinson,  Ed.  Clark,  Malberg,  Morin,  Styffe,  Staud- 
inger,  and  by  the  author.  Experiments  on  compression 
by  Hodgkinson,  Dulean,  the  engineers  of  the  Cincin- 
nati Southern  Railroad  and  Bauschinger.  Experiments 
on  flexure  by  Morin,  Woehler,  Staudinger.  Experi- 
ments on  shearing  or  torsional  Moduli  by  Duleau, 
Woehler,  etc. 

III.— Deficiencies  of  continuous  girders  and  trusses.  The 
assumption  of  a  constant  moment  of  inertia.  Con- 
tinuous  trusses  should  have  but  one  web  system.  Dis- 
turbances of  strains  from  manufacture,  settling  of  ma- 
sonry and  foundations. 

Building  continuous  girders  on  false  works  in  point  of 
quality  preferable  to  pushing  them  over  the  piers. 

Disturbances  in  strains  of  continuous  bridges  by  un- 
equal expansion. 

Continuous  bridges  require  the  best  class  of  substruc- 
ture. 


The  change  in  the  strains  of  continuous  bridges  from 
tension  into  pressure  must  be  provided  for  by  proportion- 
ing the  members  to  the  sum  of  both  strains.  Different 
systems  of  bridges  must  be  compared  by  giving  to  each 
its  proper  and  most  advantageous  proportions,  but  not 
by  supposing  all  to  be  of  the  same  depth. 

Continuous  plate  girders  not  only  are  really  more 
economical  than  single  span  plate  gilders,  but  the  the- 
ory also  is  more  in  conformity  with  the  latter  mode  of 
construction.  Many  French  engineers,  in  accordance 
with  the  supposition  of  theory,  applied  uniform  cross- 
sections  of  flanges. 

Economy  of  such  girders,  with  constant  flanges,  by 
lowering  of  middle  supports,  or  in  case  of  three  and  more 
spans  by  suitable  proportions  of  spans. 

Economy  not  enhanced  by  these  methods  if  the  chords 
are  properly  varied. 
IV.— Development  of  the  theory,  without  higher  calculus. 

Deflection  of  a  conti  lever  beam. 

"   truss  resting  freely  on  two  supports. 

Henry  Bertot's  equation. 

Correction  for  unequal  heights  of  support. 

"       expansion,  of  chords. 

Deflections  of  trusses  by  considering  the  webmembers 
as  well  as  the  chords. 

V.— Example  of  two  continuous  spans  and  single  spans. 
Influence  of  each  separate  panel  load.  Consideration 
of  excess  load  of  .locomotive.  Maxima  chord  and  web- 
strains. 

A  simple  mode  of  calculation  of  single  spans. 

Discussion  of  weights. 

Theoretical  quantities  of  single  and  of  continuous 
spans  of  two  and  more  spans.  Influence  of  dead  load. 
No  theoretical  saving  of  continuous  trusses  over  single 
spans  of  most  economical  depths. 

The  saving  in  chords  neutralized  by  loss  in  the  webs 
of  continuous  trusses. 

Correction  of  strains  on  account  of  deflections  by 
webs.  Error  to  amount  to  29  per  cent. 


VI.— Possible  irregularities  of  the  strains  of  the  two  calcu- 
lated continuous  bridges. 

Influence  of  heights  of  support,  and  of  unequal  expan- 
sion of  chords.  Modes  of  adjustment. 

Experience  as  to  influence  of  heat  of  sun  observed  on 
the  Tarascon  bridge  in  France,  the  Tfceis  bridge  in  Hun- 
gary, South wark  stone  bridge,  Britannia  and  Victoria 
bridges  in  England  and  Canada.  Drawbridges  in  this 
country  as  observed  by  Mr.  0.  Shaler  Smith  and  by  Mr. 
John  Griffen. 

Diminution  of  deflection  of  continuous  trusses  of  no 
practical  moment. 

Length  of  continuous  bridges  limited  by  longitudinal 
expansion  and  contraction. 

Deflections  of  continuous  bridges  form  a  test  of  their 
strains  being  properly  calculated  or  agreeing  with  real- 
ity. 

This  quality  recognized  by  learned  French  engineers. 

Difference  of  calculated  and  actual  deflections  pro- 
portional to  difference  of  calculated  and  actual  reac- 
tions. 
VII.— RECAPITULATION  AND  CONCLUSIONS. 

Continuous  draws,  first  investigated  by  Professor 
Stenberg  in  1857,  who  also  in  his  lectures  gave  the  idea 
of  weighing  the  reactions. 

C.  Shaler  Smith,  the  first  engineer  who  investigated 
and  built  continuous  draws  without  end  reactions  un- 
der dead  load. 

Continuous  draws  to  be  calculated  as  two  continuous 
spans. 

Drawbridges  consisting  of  two  single  spans  to  be  uni- 
ted for  each  movement  and  severed  if  in  place. 

Large  spans  with  limited  depths  can  be  built  as  con- 
tinuous trusses  with  hinges  in  alternate  span. 

These  trusses  first  patented  by  De  Bergue  in  England 
in  1865. 


PEEFAOE. 


A  paper  by  the  author,  being  a  critical  ex- 
amination into  the  merits  of  continuous  gird- 
ers, was  prepared  six  years  ago,  but  its  pres- 
entation to  the  American  Society  of  Civil 
Engineers  wa^s  delayed  till  the  last  spring. 
In  this  paper  the  subject  for  the  first  time  is 
found  to  be  based  without  the  use  of  higher 
calculus  on  one  simple  geometrical  relation, 
forming  the  connecting  link  between  single 
spans  and  continuous  bridges. 

The  same  paper,  increased  with  further 
data  resulting  from  its  discussion,  is  compiled 
into  the  short  treatise  now  presented. 

It  is  hoped  that  it  may  contribute  to  clear 
opinions  as  to  the  real  merits  of  the  systems 
of  single  and  continuous  spans,  and  would 
lead  to  a  more  thorough  understanding  of  the 
nature  of  each. 

The  subject  having  never  before  been  treat- 
ed in  this  light,  it  is  believed  that  railroad 
engineers  will  not  unfavorably  receive  the 
results  of  special  studies  which  have  occupied 


vra 

a  period  of  many  years,  and  which   in   the 
main  are: 

That  in  addition  to  the  sensitiveness  of 
continuous  bridges,  the  economy  claimed  for 
them  does  not  exist  either  theoretically  or  prac- 
tically in  all  instances  in  which  the  construc- 
tion of  properly  designed  compound  single 
span  trusses  is  not  limited  as  to  their  depths. 
As  a  result  of  these  conclusions,  there  would 
seem  to  be  one  more  good  reason  that  the 
most  valuable  time  of  polytechnic  students 
should  not  be  unnecessarily  wasted  by  enter- 
ing deeply  into  a  theory  which  more  essen- 
tially is  of  mathematical  and  historical  in- 
terest. 

C.  BENDER,  C.  E. 
Member  American  Society  of  Civil  Engineers. 

NEW  YORK,  October  12,  1876.      . 


APPLICATION  OF 

The  Theory  of  Oob^feadus  -Girders 


TO  ECONOMY  tt 


Lately  the  introduction  in  this  country  of 
continuous  girders  has  been  suggested  on  the 
plea  of  greater  economy  than,  it  is  asserted, 
can  be  obtained  under  application  of  the 
highly  perfected  and  simple  Anferican  truss- 
es. The  majority  of  advocates  of  that  sys- 
tem who  are  now  devoting  their  time  to  the- 
ory, though  well  acquainted  with  the  mathe- 
matical part  of  the  subject,  have  omitted 
some  practical  bearings  which  would  be  nec- 
essary to  enforce  their  assertions.  In  reality, 
mathematical  investigations  of  the  subject  of 
continuous  girders  require  not  a  very  high 
degree  of  training  in  analysis.  It  needs  but 
the  execution  of  the  integration  of  one  single 
equation,  which  execution  may  become  leng- 
thy and  tedious,  and  may  require  much  pa- 
tience. But  once  the  one  mathematical  idea 


10 


of  that  equation  be  understood,  the  rest  of 
the  work  is  common  and  mechanical  alge- 
braic labor. 

If  it  wei*e  tPiie  £hat  continuous  girders  give 
more  economy  thari  the  system  in  use  in  the 
C'mt^ct  States,  it;  would  rcertainly  be  a  heavy 
charge  against  the  engineers  who  have  made 
the  art  of  bridge  building  their  specialty,  and 
who  study  their  profession  with  all  earnestness. 
Many  will  deny,  at  the  outset,  that  this  charge 
is  just ;  for  the  sake  of  others,  the  author  pro- 
poses to  show  that  the  American  practice  of 
bridge  building  hitherto,  has  been  in  the  prop- 
er direction*towards  other  improvements,  and 
that  the  theorists  who  wish  bridge  builders 
to  follow  their  advice  have  studied  the  sub- 
ject in  but  one  of  its  bearings,  and  have 
omitted  to  examine  closely  their  premises  as 
well  as  their  conclusions. 

The  author  believes  it  is  not  only  desirable, 
but  necessary  that  this  question  should  be  fully 
discussed,  from  various  reasons.  Practical  en- 
gineers generally  do  not  place  much  confi- 
dence in  long  formulas,  and  if  they  once  have 
studied  mathematics  thoroughly  they  lose 
the  taste  for  these  studies  after  some  time  of 
practice,  since  they  have  convinced  them- 


11 


selves  as  to  the  futility  of  ultra  refined  theo- 
retical speculations.  These  engineers  will 
not  be  very  likely  to  adopt  structures  whose 
calculation  of  strains  would  waste  so  much 
valuable  time.  But  these  engineers  could 
not  prevent  a  new  method  of  construction  in 
time  becoming  fashionable,  whether  correct 
or  not,  as  long  as  it  were  founded  on  some 
elegant  theory  and  seemingly  led  to  econ- 
omy. For,  under  our  large  factors  of  safety, 
we  can  commit  many  sins  in  construction  be- 
fore they  are  found  out.  Again,  there  is 
always  a  number  of  men  who,  because  they 
do  not  understand  abstruse  calculations  and 
formulae,  rather  than  admit  this  fact,  publicly 
endorse  them  warmly.  And  finally,  when  in 
polytechnic  schools  for  a  number  of  years,  a 
certain  theory  has  been  thoroughly  studied 
with  zealous  assiduity,  a  little  army  of  its 
admirers  will  fill  positions  in  railroad  and  in 
public  engineering  offices,  anxiously  waiting 
for  the  first  opportunity  towards  introducing 
into  practice  what  they  consider  the  finest 
jewel  of  their  technical  knowledge.  The  au- 
thor frankly  admits  once  to  have  been  of  this 
number.  But  after  studying  the  subject  of 
continuity  of  trusses  for  several  years,  and  a 


12 


careful  examination  of  its  suppositions  he  found 
himself  compelled  to  admit  that  the  theory  is 
not  correct  scientifically,  and  does  not  agree 
with  the  physical  laws  of  elasticity  of  iroif.* 

We  are  now  prepared  to  prove,  that  for  me- 
dium spans,  say  of  200  feet,  the  construction 
on  the  principle  of  continuity  leads  to  greater 
truss  weights  in  addition  to  greater  cost  of 
workmanship  than  are  required  by  the  use  of 
single  spans  with  improved  details. 

This  last  result  is  very  important  indeed, 

*  Six  years  ago,  in  a  paper  written  for  the  German  Soci- 
ety of  Engineers,  Verein  JDeutscher  ingenieure  in  Berlin, 
which  was  translated  into  English  and  published  two  years 
ago  in  the  Railroad  Gazette  of  New  York,  the  author  stated : 

44  The  writer  of  these  lines  himself  had  for  some  time 
thought  that  it  might  be  possible,  by  application  of  pin 
joints,  by  reducing  the  number  of  parts,  by  the  use  of  proper 
scales  and  adjustments  for  the  regulation  of  the  pressures 
on  the  three  or  more  piers  of  a  continuous  bridge,  and  by  the 
use  of  scientifically  correct  and  complete  formulae,  to  pro- 
duce reliable  continuous  trusses,  by  means  of  which  the 
large  rivers  of  this  country  could  bespanned  without  the  use 
of  false  works." 

"With  a  great  deal  of  labor  he  had  constructed  an  analyti- 
cal expression,  which  embraced  the  relation  of  the  moments 
of  flexure  over  three  consecutive  piers  of  a  continuous  gird- 
er. In  this  formula,  due  attention  was  given  not  only  to  the 
deflections  caused  by  the  chords,  but  also  to  those  due  to 
the  tensile  and  compressive  members  of  the  web  system  ; 
also  the  actual  section  of  each  separate  member  was  intro- 
duced. It  therefore  did  away  with  two  errors  of  the  formu- 
lae generally  quoted  in  books,  which  are  only  applicable 
when  the  girders  are  very  shallow  and  when  the  web  is  a 


13 


for  if  it  were  possible,  under  application  of 
the  principle  of  continuity  to  arrive  at  an 
economy  in  weight  and  cost,  there  would  be 
a  large  market  for  this  article  however  objec- 
tionable the  mode  of  construction;  for,  rail- 
road officers  in  the  majority  of  instances  will 
be  led  by  the  consideration  of  first  cost;  es- 
pecially since,  in  bridge  building  (thanks  to 
our  factor  of  safety)  many  errors  remain  un- 
punished for  a  long  time,  continuous  girders 
with  their  delusive  theory  and  deceptive  stiff- 
ness under  application  of  lattice  and  rivets 
would  gain  a  wide  market. 

plate,  and  which  even  under  those  suppositions  do  not  coin- 
cide very  satisfactorily  with  experiments." 

u  Notwithstanding  the  theoretical  improvements  men- 
tioned, it  was  finally  found  that  the  labor  spent  in  finding 
said  formula  had  been  in  vain,  from  a  reason  which  in  Eu- 
rope, as  far  as  known,  has  not  received  any  consideration. 
It  is  the  great  variability  of  the  modulus  of  elasticity,  which 
in  the  formulae  of  the  books  is  supposed  to  be  a  constant 
value  of  about  25,000,000  pounds  per  square  inch." 

"  But  the  writer  has  tested ,  during  his  presence  at  the 
Phoenix  Iron  Works,  many  thousands  of  eye-bars,  made  for 
actual  use  in  bridges,  and  found  that  the  modulus  of  these 
members  is  very  changeable,  namely  from  18,000,000  to  over 
40,000,000  pounds  per  square  inch,  so  that  small  sections  give 
the  lowest  and  large  sections  the  greatest  figures.  The  same 
result  was  obtained  by  the  Canadian  engineer  who  inspect- 
ed the  iron  for  the  International  bridge  near  Buffalo,  as  well 
as  by  Mr.  B.  Nicholson,  who  was  sent  to  Phoenixville  by  the 
government  officers  of  the  United  States  to  inspect  the  iron 
for  the  Mississippi  bridge  at  Rock  Island." 


14 


The  theory  of  continuous  girders,  as  given  in 
text-books,  does  not  always  permit  the  philoso- 
phy of  the  principle  involved  to  be  clearly  seen : 
its  representation  generally  is  rather  obscure. 
In  order  to  explain  this  principle  as  clearly  as 
possible,  the  author  worked  out  a  new  method 
of  treating  the  subject.  The  results  under 
this  treatment  naturally  must  agree  with 
those  derived  from  the  application  of  the 
general  theory  of  the  elastic  line  which,  in 
the  last  century  (1744),  was  first  given  by 
Leonard  Euler  of  Basel,  then  member  of  the 
Academy  of  Science  in  Berlin,  which,  by 
Navier,  early  in  this  century,  was  propagated 
among  engineers,  and  lately  was  somewhat 
simplified  by  Henry  Bertot  in  France.* 


*  Jacob  Bernoulli!  having  in  the  year  1695  given  ike  no- 
tion of  the  "  neutral  line,"  tried  in  1705,  shortly  before  his 
death,  to  find  the  equations  and  properties  of  the  "  elastic 
line."  In  this  he  did  not  succeed,  but  Leonard  Euler  in  his 
book  "de  curvis  elasticis"  (Lausanne  and  Geneva),  1744, 
solved  this  problem,  showing  that  for  flat  elastic  curves  the 
second  differential  co-efficient  is  proportional  to  the  moment 
of  flexure  of  exterior  forces.  P.  S.  Girard,  in  his  work 
"  Traite  analytiqiie  de  la  resistance  des  solides,  1798,"  page 
50,  &c.,  translated  Euler's  treatise  from  the  Latin  into  the 
French  language,  and  he  adds  as  an  application  of  Euler's 
theory  the  investigation  of  a  beam  fixed  at  both  ends.  Ey- 
telwein  and  Navier  extended  this  labor  to  the  beam  continu- 
ous over  three  and  more  supports.  Henry  Bertot  in  France, 
in  the  year  L855  (Comptes  Rendus  de  la  societe  des  logo- 


15 


I. — THE    GENERAL    PRINCIPLE    INVOLVED   IN 

THE  THEORY  OF  CONTINUOUS  GIRDERS. — We 
first  consider  a  number  of  single  spans  of  the 
lengths  £1?  4>  4>  4?  touching  each  other  re- 
spectively over  the  piers  B>  C,  D,  E.  We 
suppose  each  span  to  be  loaded  in  any  con- 
ceivable or  desired  manner  ;  in  consequence, 
each  span  would  deflect  so  as  to  form  certain 
curves  as  indicated  by  dotted  lines.  The 
lower  chord  would  not  remain  straight,  the 
end-posts  would  not  remain  vertical.  Differ- 
ing with  the  nature  of  the  material  with  the 
sectional  areas  of  the  members  of  the  bridge, 
with  the  loads  imposed  upon  them,  the  trusses 
would  show  certain  angles  yiy  o\,  j/2,  #2,  ;/„, 

nienrs  Civils  cle  Pans,  page  278,  &c.),  tor  the  iirst  time  gave 
what  is  called '•  the  theorem  of  the  three  moments,"  which 
later  (1857),  and  independently  of  Bertot,  was  found  by  Cla- 
peyron  and  Bresse  in  France,  and  by  the  English  engineer, 
Heppel,  in  the  year  1858,  in  India. 

In  the  United  States  Colonel  Long,  the  well  known  in- 
ventor, in  the  boos  on  his  patent  (1838)  truss,  edited  in  1841 
in  Philadelphia,  probably  for  the  first  time  has  applied  the 
principle  of  continuity  to  wooden  skeleton  trusses  and  num- 
bers of  such  wooden  bridges  since  then  were  built  in  this 
country. 

Mr.  Pole  in  1852  applied  the  theory  to  a  bridge  over  the 
Trent,  consisting  of  two  continuous  spans  of  130  feet  each, 
and  this  engineer  also  did  the  analytical  part  of  the  calcula- 
tion of  the  strains  of  the  Britannia  bridge,  such  as  contained 
in  Mr.  Edwin  Clark's  famous  work.  (Vide  Transactions  I. 
C.  Engineers,  Vol.  xxix.) 


to 


dg,  &c.,  <fcc.,  of  the  end-posts  with  their 
originally  vertical  positions.  Now,  suppose 
you  draw  together  the  top  points  of  the  end- 
posts  over  the  piers  B,  (7,  Z>,  E,  and  press 
apart  the  bottom  joints  of  these 
posts,  so  that  not  only  the  top 
but  also  the  bottom  chords  of 
the  adjacent  spans  would  touch 
each  other;  or,  in  other  words, 
insert  certain  tensile  forces  in- 
to the  upper  chord,  and  equally 
large  compressive  forces  into 
the  lower  chord  of  each  truss. 
Thus  each  truss  by  a  certain 
unknown  moment  of  flexure 
would  artificially  be  bent  up- 
ward in  such  a  manner  that 
certain  angles  al9  /?,,  ^2,  /?2, 
av  fis?  would  be  produced, 
were  the  dead  and  the .  live 
loads  of  the  trusses  removed. 
When  at  each  central  pier, 
the  desired  continuity,  consist- 
ing of  connection  of  the  top 
and  bottom  chord  ends,  separ- 
ated under  the  dead  and  live 
loads  alone,  but  overlapping  each  other  un- 


^--- 


der  the  action  of  the  mo- 
ments M19  MV  M3,  <fcc.,  is 
effected,  this  law,  must  ob- 
tain, namely:  the  sum  of 
the  angles  of  deflection  at 
any  central  pier  caused  by 
the  (dead  and  live)  loads  on 
two  adjacent  trusses  must  be 
equal  to  the  sum  of  the 
angles  of  elevation,  caused 
by  the  unknown  moments 
artificially  applied  at  the 
three  piers  of  the  contempla- 
ted spans.  This  is  expressed 
algebraically  : 


&o.   3 

In  these  equations  the  left 
sides  are  functions  of  the 
dead  and  live  loads,  and  the 
right  of  the  unknown  mo- 
ments, MV  M2,  MV  &c. 

For  each  intermediate  pier  there  is  one 
equation  and  one  unknown  moment.  The 
number  of  equations  equals  the  number  of 
unknown  moments,  which  equals  the  number 


18 


of  spans  of  the  continuous  bridge,  less  one. 
For  n  spans  there  are  (w~— l)  equations  and 
(n — 1)  unknown  moments.  These  (n — l) 
equations  are  therefore  sufficient  to  solve  the 
problem  which  is  identical  with  finding  the 
unknown  moments.  From  the  theory  of  sin- 
gle spans,  which  again  is  founded  only  on 
the  law  of  the  lever,  we  calculate  angles  of 
deflection  like  6,  y,  a^  //,  and  therefore  the 
above  law,  expressed  by  the  very  plain  equa- 
tions (7"),  indicates  how  to  derive  the  strains 
of  continuous  girders  from  those  of  single 
spans.  This  law,  as  it  were,  is  the  tune  for 
all  the  rest  of  variations  relating  to  continu- 
ous girders. 

If  we  express  the  angles  a  and  /?,  by  the 
unknown  moment  acting  on  two  continuous 
spans  we  arrive  directly  at  the  formula  of 
Henry  Bertot,  improperly  ascribed  to  Cla- 
peyron,  which  is  found  after  a  tedious  pro- 
cess of  integration.  Bertot 's  formula  in  fact 
expresses  only  the  geometrical  law  that  the  sum 
of  the  angles  of  deflection  must  be  equal  to  the 
sum  of  elevations  due  to  the  moments  M^  M^ 
J/3,  cfcc.,  over  the  central  piers. 

Though  we  suppose  the  reader  to  be  ac- 
quainted with  the  theory  of  single  span 


19 


bridges,  in  the  sequel  we  shall  develop  a  few 
rules  belonging  to  this  theory  sufficient  for 
us  to  directly  write  down  the  formula  from 
which  we  calculate  the  moments,  M19  M^ 
c%c.,  and  consequently  the  strains.  Before 
doing  this  we  first  wish  to  more  fully  explain 
the  next  consequence  of  the  principle  of  con- 
tinuous girders.  A  span  A  B  ;  without 
weight  is  resting  on  two  supports  A  and  B; 
at  B  a  moment  Ml  equal  to  forces/,—;/,  with 
the  lever  h  acts  on  the  chords.  This  moment 
is  counteracted  by  a  force  (weight)  />,  hold- 
ing the  truss-end  A  to  the  pier.  Though 

Fi£,  3. 


/      


the  force  />,  holds  down  the  end  A,  yet 
the  moment  M^—fli  will  cause  a  convex 
elastic  curve,  so  that  the  end  posts  which 
originally  were  vertical,  are  made  to  form 


20 


angles  a  and  /?,  to  their  vertical  positions. 
Since  a  moment  of  flexure  can  only  be  neu- 
tralized by  another  opposite  moment,  there 
must  exist  another  force, — pl9  acting  on  the 
pier  .Z?,  which  in  "combination  with  -\-p^  on 
the  lever  Z,  equals  precisely  Ml=.fh9  in  other 

M 

words,  />!  must  be  equal  to  -;-  and  M^=pJ,r 

*i 

For  any  section  (7,  of  the  beam  A  J3,  the 
moment  of  flexure  is  equal  to  the  force  pl9 
multiplied  by  the  distance  x,  and  the  greater 
x  is,  the  greater  the  moment  of  flexure  in  C. 
When  x  becomes  equal  to  119  the  maximum  of 
the  moment  is  reached,  namely  J/j=^)1  ^, 
whilst  at  A,  the  moment  of  flexure  acting  on 
the  beam  is  zero,  because  x  is  zero.  And  as 
the  curvature  of  a  beam  increases  directly  as 
its  moment,  the  beam  is  not  bent  at  all  at  A, 
but  is  gradually  bent  more  and  more  the 
nearer  we  come  to  J3. 

From  what  has  been  said,  this  law  can  be 
deduced:  that  the  application  of  a  moment 
Mv,  to  the  central  pier  of  an  end  truss  span  of 
a  continuous  girder  does  call  forth  two  forces 
•+•/?!>  — p^  which,  however,  do  not  alter  the 
sum  of  the  reactions  A  and  IB  of  this  span  in 
whatever  manner  it  may  be  loaded. 


21 


The  moment  M,  reduces  the  pressure  on 
the  pier  A,  but  only  by  increasing  with  the 
same  amount — pl9  the  pressure  on  the  pier  B. 
From  this  observation  it  further  follows  that 
by  the  principle  of  continuity,  no  load  rest- 
ing on  an  end  span  can  be  carried  over  to  the 
next  span,  but  that  the  sum  of  these  loads 
always  is  neutralized  by  the  two  nearest  piers 
between  which  it  acts.  The  distribution  only 
of  the  reactions,  which  for  single  spans  is 
governed  by  the  law  of  the  lever,  in  the  end 
spans  of  continuous  girders  is  modified. 

What  has  been  said  of  a  single  span  acted 
upon  by  one  moment  Miy  is  equally  true,  if 
on  its  other  end  another  moment  Jf2,  would 
act.  All  we  need  do  is  to  add  together  the 
effects  due  to  each  separate  moment.  There- 
fore, also,  any  load  acting  on  a  middle  span 
of  a  continuous  girder  is  taken  up  by  the  two 
nearest  piers.  Also,  in  this  instance,  the 
sum  of  the  two  partial  reactions  belonging 
to  this  span  on  these  piers,  equals  the  total 
load  between  them.  Only  the  proportion 
between  these  reactions,  by  the  principle  of 
continuity  is  modified. 

This  result  could  have  been  anticipated 
from  the  following  consideration :  The  strut 


22 
C  D,  Fig.  4,  carries  down  to  the  pier  D  the 


loads  due  to  the  span  G.  As  long  as  the 
bearing  D  is  inelastic,  the  diagonal  D  E  can 
not  be  drawn  down  and  the  vertical  pressures 
carried  by  the  members  C  D  and  E  D  must 
be  directly  annihilated  in  D.  The  case  would 
be  very  much  different  if  the  pier  D  were 
elastic,*  for  then  there  would  arise  a  deflec- 
tion of  the  truss  G  Fin  D,  and  a  portion  of 
the  shearing  force  from  one  truss  could  travel 

o 

to  the  next  one. 

Having  learned  that  the  moments  M^  J/2, 
J/3,  cause  the  existence  of  pairs  of  forces 

+  Pi  —  Pv  +P»  —P»  +  PB  —  Pv  +  P*  —  Pv 
<fec.,  it  is  very  easy  now  to  express  the  exact 
value  of  the  moments  by  the  forces  pl9  p^ 

*  The  supposition  of  elastic  supports,  consisting'  of  systems 
of  springs,  was  investigated  by  the  writer  in  1867,  in  an  ex- 
tensive series  of  calculations,  with  a  view  to  determine  whe- 
ther thereby  any  economy  could  be  secured,  and  with  the 
intention  to  use  these  springs  as  adjusting  and  weighing  ap- 
paratus of  the  elastic  reactions.  The  result  was,  that  the 
variable  positions  of  the  movable  load  so  much  reduced  any 
gain  in  the  chords  that  the  additional  expense  of  the  systems 
of  springs  left  no  economy  for  a  structure  of  this  kind. 


<fcc.,  and  their  levers,  119  12,  1Q9  &c.  These 
values  simply  are  (n  being  the  number  of 
spans) : 

^  and,  finally,    0  = 

^  M«.  —  sum  of  m°- 

ments  (pi)— 2 (pi), 
(from/?!  top,).  The 

last  moment  Mn.^  is 
equal  to  2n.l  (pi), 
and  also  is  equal  to 
the  moment />„  /„,  so 
that  the  (n-l)  mo- 
ments are  sufficient 
to  calculate  the  n 
forces  ^/^  .  .  .  pn. 
After  these  prepar- 
ations we  proceed 
to  the  values  of  the 
angles  a\  _ftl  yl  6l ; 

0/2  P2  Y2  °V2  5  ^C<5  &C> 

s^  ^  §^    Can     these    values 
f»<r    "*    _•* 

%H 

^- 


w 


V 


^' 


really  be  calcula- 
ted ?  With  the  an- 
swer of  this  ques- 
tion in  the  affirma- 
tive or  negative 
stands  or  falls  the 


24 


whole  theory  of  continuity  of  girders;  and 
this  question  is  followed  by  another:  can  we 
calculate  with  a  sufficient  degree  of  reliabili- 
ty the  elastic  line  of  a  single  span  bridge  ? 
The  originally  straight  truss  A  jB  through 

Fig.  6. 


the  influence  of  live  or  dead  loads  has  re- 
ceived a  deflection.  The  angle  A  C  B,  which 
originally  equalled  180°,  has  been  altered, 
and  the  alteration  x  C  J3  was  caused  by  the 
loads. 

The  question  then  arises,  can  the  angle 
xCB  be  calculated  with  a  sufficient  degree 
of  reliability  ?  The  alteration  of  the  angles 
around  C  is  equal  to  the  sum  of  the  altera- 
tions of  the  separate  angles.  Each  angle  is 
altered  because  each  side  of  each  triangle  has 
been  altered  in  length.  Some  sides  have 
been  shortened  under  pressure,  some  have 
extended  under  tension.  These  extensions 
and  compressions  are  very  small,  and  it  is 
very  difficult  to  measure  them. 

If  we  knew  the    alterations  of  the  sides, 


25 


also  hence  the  altered  angles,  calculation 
would  be  possible.  But  to  determine  the  ex- 
tension or  compression  of  each  side  of  each 
triangle,  it  is  necessary  to  know  for  each  side, 
the  total  strain  and  the  exact  value  of  the 
cross-section,  and  precisely  how  much  each 
side  (each  member)  will  extend  or  compress 
under  the  action  of  a  ton  per  square  inch. 
The  extension  or  compression  of  a  member  of 
a  girder  therefore  depends  on  the  strain  per 
square  inch  at  any  point  of  this  member,  and 
on  quality  of  the  material. 

For  plain,  single  span  bridges  with  hinged 
joints,  the  law  of  the  lever  teaches  us  to  cal- 
culate the  total  strain  in  any  member.  We 
have  reduced,  in  principle,  the  problem  of 
continuous  girders  to  a  combina.tion  of  prob- 
lems on  single  spans.  Hence  also  for  con- 
tinuous girders  we  could  determine  those  to- 
tal strains,  if  it  were  possible  to  calculate  the 
deflections  of  single  span  girders.  But  we  do 
not  ki.ovv  beforehand  the  sectional  areas  of 
the  different  members  of  a  continuous  girder; 
on  the  contrary,  it  is  just  our  problem  to  find 
those  sections  which  are  most  suitable. 

The  theory  of  continuous  girders  as  treated 
in  text-books,  leaps  over  this  difficulty  by 


26 


making  an  arbitrary  supposition,  namely, 
that  all  sections  are  equal.  We  shall  have 
occasion  to  show  that  this  is  theoretically  in- 
correct, and  causes  an  error  of  as  high  as  15  per 
cent,  of  the  calculated  values,  and  this  error 
probably  is  as  large  as  the  theoretical  gain 
claimed  for  the  chords  of  continuous  girders. 

As  far  as  the  nature  of  the  material  is  con- 
cerned, we  know  that  extensions  and  com- 
pressions are  proportional  to  the  strains  per 
square  unit,  and  that,  in  order  to  find  them, 
the  strain  per  square  unit  must  be  divided  by 
a  coefficient  proper  to  the  material  which  is 
called  the  modulus,  and  the  quotient  thus  ob- 
tained must  be  multiplied  by  the  original 
length  of  the  member.  The  next  question  is, 
therefore,  do  we  know  the  value  of  this  mod- 
ulus for  the  material  used  ? 

This  question  must  be  answered  by  experi- 
ment. It  belongs  to  the  science  of  natural 
philosophy.  The  physical  suppositions  upon 
which  mathematical  investigations  are  based 
should  always  be  founded  on  undeniable  facts, 
since  the  truth  of  these  suppositions  is  the 
Qonditio  sine  qua  non  of  the  value  of  the  re- 
sulting formulae.  For  the  supposition  being 
reduced  to  a  mere  hypothesis,  it  is  wholly 


indifferent  by  what  brilliant  and  elegant  ana- 
lytical or  graphical  method  the  deficiencies 
of  the  foundations  are  hidden. 

The  theory  assumes  that  the  modulus  of 
elasticity  for  any  part  of  a  girder  is  a  con- 
stant value  unaltered  to  any  noticeable  extent 
by  manufacture ;  or  by  rivets,  covering  plates, 
different  sections  of  rolled  iron,  or  by  thick- 
ness of  the  metal,  <fec.,  &c.  Practical  men 
will  be  likely  to  demand  that  the  truth  of  this 
broad  hypothesis  be  demonstrated.  There- 
fore, previous  to  calculating,  let  us  examine 
the  hypothesis  of  the  theory  of  continuity. 
Nowhere,  in  applied  science,  the  necessity  for 
such  examination  is  more  urgent. 

The  extensions  and  compressions  allowed 
in  practice  are  very  small  quantities.  De- 
fective testing  machines,  •unacquaintance  or 
unfitness  of  experimenters  to  such  delicate 
work,  temperature,  variability  of  manufac- 
ture of  material  under  test,  variability  of  the 
chemical  composition,  density,  uniformity, 
&c.,  are  causes  of  great  errors.  Often  the 
number  of  tests  were  too  small  to  draw  there- 
from any  justified  conclusion,  or  the  elements 
of  time  and  motion  have  been  neglected,  or 
experimenters  overlooked  other  important 


28 


elements  altogether.  There  are  also  instances 
that  experimenters  were  not  impartial,  and 
that  theories  have  been  formed  first,  and 
experiments  have  been  arranged  afterward 
to  suit.  Experimenters  may  reject  results 
which  in  their  opinion  seem  untrustworthy, 
while  they  report  and  elaborate  others  which 
to  them  seem  probable  because  favorable  to 
their  theory.  Or  experiments  were  made  on 
one  sort  of  material  and  applied  to  material 
of  quite  a  different  nature  and  section.  It  is 
by  no  means  an  easy  labor  to  conduct  trust- 
worthy experiments  on  the  elasticity  of  mate- 
rial, and  Professor  Wullner*  is  perfectly  cor- 
rect in  saying:  "  The  examination  of  the  elas- 
ticity of  solid  bodies  is  one  of  the  most  diffi- 
cult in  the  whole  science  of  natural  philoso- 
phy. In  order  to  .conceive  its  laws  sufficient- 
ly, the  most  intricate  mathematical  investi- 
gations and  the  most  subtle  experiments  are 
required."  These  two  conditions  rarely  are 
found  combined. 

We  shall  soon  see  'that  the  theory  of  con- 
tinuous girders  was  built  up  exclusively  by 
men  of  purely  mathematical  capacities,  and 


*  In  the  first  volume  of  his  work  on  physics. 


29 


that  they  did  not  begin  with  that  cautious 
examination  of  their  suppositions  which  is 
demanded  by  true  science  as  much  as  by 
practice.  It  is  true  that  some  theories  have 
been  worked  out  before  the  basis  properly 
was  investigated,  and  are  applicable  because 
the  results  of  such  theories  were  finally  tested. 
But  we  have  no  such  experiments  on  contin- 
uous trusses.  For  the  measured  deflections 
of  executed  continuous  bridges  are  neither 
sufficiently  exact  nor  are  they  of  use  for  our 
purpose,  since  we  do  not  know  what  per- 
manent sets  were  produced  after  removal 
of  the  false-work,  nor  was  the  modulus 
of  each  member  previously  examined  and 
recorded. 

This  remark  also  applies  to  single  spans,  and 
the  question  therefore  again  most  pertinently 
arises,  whether  the  exactly  calculated  deflec- 
tion of  a  single  span  ever  did  agree  with  its 
real  deflection.  Logic  compels  us  to  acknow- 
ledge that  such  coincidence  is  impossible, 
save  by  sheer  accident  ;  for,  who  has  ex- 
amined the  modulus  of  each  finished  riveted 
compression  member,  or  of  each  rod  or  bar 
of  a  bridge  ?  Who,  therefore,  was  capable 
of  calculating  the  deflections  ?  ]\Ioreover, 


30 


these  calculations  generally  have  been  faulty,* 
and  if  engineers  assert  that  the  deflections  of 
their  bridges  agree  with  their  calculations, 
either  the  first  were  erroneously  calculated  or 
the  second  not  well  observed. 

Since  we  have  no  experiments  on  the  quali- 
ties of  finished  continuous  girders,  the  more 
should  we  examine  the  results  of  experiments 
on  the  values  of  moduli  of  iron  and  steel. 
These  experiments,  fortunately,  are  very  nu- 
merous indeed,  and  gives  us  all  information 
wanted,  whilst,  unfortunately,  theorists  on 
continuity  have  not  considered  it  worth  while 
to  consult  this  great  quantity  of  scientific 
material  previous  to  entering  into  mathe- 
matical speculations. 

II.  EXPERIMENTS  ON  THE  VALUES  OF  THE 
MODULI  OF  IRON  AND  STEEL. — Bornet,  in 
France,  over  40  years  ago,  made  experiments 
on  rods  of  chain  iron  0.2  inches  in  diameter 

*  The  influence  of  the  webs  on  deflections,  generally  is 
neglected.  With  the  exception  of  by  Sclrwedler  (see  official 
Engineering  Periodical— Zeitshcrift  fur  Bauwesen,  Berlin], 
no  successful  attempt  was  made  to  examine  the  influence  of 
web  posts  and  diagonals.  In  the  sequel  we  shall  show  that 
this  influence  is  enormous  and  very  perplexing ;  in  fact,  that 
calculations  of  continuity  of  bridges,  without  properly  con- 
sidering the  webs,  are  worse  than  worthless.  The  calcula- 
tions there  shown  were  first  developed  by  the  writer  in 
1869,  and  are  not  found  elsewhere. 


81 


and  21  feet  long.  The  original  modulus 
found  was  35,500,000  pounds,  and  under 
strains  of  20,000  per  square  inch,  it  decreased 
to  28,500,000  pounds.  Ardant,  in  France, 
for  soft  annealed  wire,  up  to  strains  of  35,000 
pounds  per  square  inch,  found  the  modulus 
to  be  24,000,000  pounds,  and  for  hand-drawn 
wire,  up  to  strains  of  42,000  pounds  per 
square  inch,  it  was  27,300,000  pounds.  Whilst 
of  wire,  Ardant  did  not  perceive  any  lower- 
ing of  the  modulus  up  to  strains  of  42,000 
pounds,  Bornet  remarked  a  diminution  be- 
ginning with  strains  of  8,500  pounds  per 
square  inch.  It  is  not  stated  by  Morin,  who 
reports  these  results,  in  what  manner  the  ex- 
periments were  made. 

Hodgkinson  made  (only)  two  tensile  exper- 
iments on  long  rods  to  determine  their  exten- 
sions with  exactness,  and  found  the  orginal 
modulus  of  one  bar  equal  to  23,900,000,  and 
of  the  other  to  22,400,000  pounds  ;  Edwin 
Clark  gives  29,000,000  pounds.  Vicat's  ex- 
periments on  hard  wire  with  0.1  inches  diam- 
eter, resulted  in  a  modulus  of  28,200,000,  and 
for  wire  well  annealed,  20,660,000  pounds. 
Morin  for  hardened  wire  gives  28,100,000, 
and  for  annealed  wire  22,400,000  pounds. 


Experiments  of  a  more  practical  value 
were  made  by  Malberg,  in  Prussia,  on  occa- 
sion of  his  building  the  Mulhelm  suspension 
bridge.  The  bars  for  this  structure  were 
made  by  Herr  Daelen.  The  iron  was  of 
best  German  stock,  the  puddle  loops  well 
hammered,  rolled,  piled,  and  re-rolled.  All 
bars  were  of  the  same  stock,  same  make, 
same  length,  same  sectional  shape  and  area. 
Their  moduli,  however,  varied  from  20,000,- 
000  to  27,000,000  pounds,  which  gives  a 
difference  of  35  per  cent,  for  the  same  kind 
and  section  of  bars.*  This  great  variability 
of  moduli  of  bars  of  even  the  same  shape  and 
material,  was  further  noticed  on  occasion  of 
the  construction  of  the  Vienna  Railroad  sus- 
pension bridge,  where  bars  of  the  same  modu- 
lus were  put  into  the  same  panels. 

The  author  had  occasion  to  test  many 
thousand  of  eyebars,  up  to  about  forty  feet 
length,  and  varying  in  section  from  1  to  14.25 
inches  square.  The  moduli  of  these  bars 
varied  much  according  to  their  cross-sections, 
arid  were  from  18,000,000  to  40,000,000 

*  Herr  Daelen  is  an  authority,  known  by  his  universal 
mill,  a  treatise  on  the  art  of  shape  rolling,  and  his  invention 
pf  weldless  rolled  eyebars,  known  as  Howard's  Patent. 


33 

pounds,  and  even  higher.  These  results  were 
confirmed  by  other  inspectors  of  bridge  work, 
for  instance,  by  Mr.  B.  Nicholson.* 

We  turn  to  moduli  of  steel.  Morin,  for 
steel  0.167  inches  square,  from  Petin  &  Gau- 
det,  found  31,000,000  and  31,800,000  pounds. 
Direct  tensile  experiments  on  Krupp's  steel, 
by  Woehler,  gave  on  the  average,  a  modulus 
of  32,560,000  pounds,  whilst  from  flexure  he 
calculates  31,100,000  pounds. 

Prof.  Staudinger,  of  Munich,  has  made 
careful  tests  on  Bessemer  metal, t  when  the 
moduli  were  found  to  be  independent  of  the 
quantity  of  carbon  combined  with  the  iron.J 
The  following  table  contains  the  results  of 
his  experiments  : 


*  At  Phoenixville,  Pa. 

t   From  the  Pernitz  Works  in  Austria. 

I  The  quantity  of  carbon  rose  from  0.14  to  0.96  of  one  per 
cent.  Metal  with  0.14  is  soft  iron,  with  0.19  to  0.30  it  is  gran- 
ular iron  (of  a  fine  grain)  or  hard  iron,  then  comes  soft  steel, 
which  increases  in  hardness  with  the  carbon  contained. 


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contains  0. 
modulus  th 


At  the  Vienna  Exposition,  a  set  of  test- 
pieces  could  be  seen,*  which  showed  as  fol- 
lows : 

Carbon— per  cent 1.      0.75  0.5     0.28     012 

Specific  gravity 7.837.84  7.85    7.86      7.88 

Tensile    modulus   (mil- 
lions pounds. .. 25.1    24.6  27.7    24.9.     26.1 

Ultimate  tensile  strength — 

90  000        80  000        70  000        67  000  65  000    pounds 

The  modulus  of  this  class  of  steel  and  iron 
was,  in  the  average,  noticably  lower  than 
those  for  Ternitz  iron  and  steel,  the  difference 
being  about  20  per  cent. 

B.  Baker  made  some,  experiments  on  steel 
bars  previous  to  his  experiments  on  crippling 
strength  and  found  the  modulus  from  29,100, 
000  to  37,330,000  pounds,  which  result  shows 
a  difference  of  28  per  cent.  He  saysf 
"  Every  practical  man  who  has  noted  the 
behaviour  of  iron  girders  under  bending 
stresses,  knows  whilst  one  girder  may  deflect 
a  certain  amount  under  the  test,  another  one 
precisely  similar  and  placed  apparently  under 
precisely  the  same  condition,  may  deflect 
some  30  per  cent,  more  or  less." 

Hodgkinson  made  two  direct  experiments 

*  Bessemer  metal  from  the  Reschitza  Works  in  Hungary. 
t  In  his  book  on  Beams  and  Columns. 


36 


on  the  compression  moduli  of  iron,  and  found 
19,200,000  and  21,000,000  pounds.  These 
two  experiments  strengthened  Hodgkinson 
in  his  belief  of  the  correctness  of  his  theory 
as  to  a  weakness  of  wrought  iron  under 
crushing  stresses,  whilst  they  only  prove  how 
easily  an  experimenter  may  be  misled.  Du- 
leau,  in  France,  directly  measured  the  com- 
pressions of  fibres  in  comparison  with  their 
extensions,  which  he,  differing  from  Hodg- 
kinson, found  to  be  exactly  equal. 

Very  valuable  hints  as  to  the  qualities  of 
iron  can  be  derived  from  experiments  on 
flexure,  which  can  be  conducted  easily  with 
sufficient  accuracy.  Morin,  by  such,  deter- 
mined the  following  moduli  : 

For  iron  from  works  near  Rouen 31,800,000  pounds. 

"      "      "      Jackson  Petin  &Gaudet...  28,400,000       u 

k.      «      u      Ale,  Lik  (Algeria) 28.960,000 

"    English  crown  bars 23,440,000 

"•    French  I  beams  with  equal  flanges. .  29,330,000       " 
»*          "  "          u      unequal    "     ..  24,400,000 

"    beams  also  from  Dupont  &  Drey- 

fussin  Ars  sur  Moselle 26,00,0000        " 

"    beams  also  from  Dupont  &   Drey- 

fussin,  equal  flanges 23,600,000       k' 

**    beams  also  from  Dupont  &  Drey- 

fussin,  unequal  flanges 23,000,000       ** 

"    beams  also  from   Dupont  &  Drey- 

fussin,  same  beam  reversed 23,000,000       " 

Here  again  we  have  differences  of  moduli 


37 


amounting  to  39  per  cent.,  and  for  the  same 
class  of  iron  (Lorraine  beams)  of  27  per  cent. 

Thomas  D.  Lovett*  has  lately  furnished  an 
elaborate  series  of  experiments  on  compres- 
sion members,  such  as  actually  used  in  the 
bridges  of  the  Cincinnati  Southern  Ry.  Up 
to  the  time  of  his  report  t  30  compression 
members  had  been  tested  and  broken;  their 
moduli  varied  from  19,300,000  to  34,600,000 
pounds.  J 

Experiments  on  hollow  wrought  iron  tubes 
made  by  Hosking  gave  these  results : 

Moduli  of  a  rectangular  tube 20.405,000  pounds 

"        "      round  tube 24,500,000       " 

"       elliptic 24,300,000       " 

Moduli  of  rails,  experiments  made  by  Mo- 

rin: 

Tredegar  iron,  double  headed,  maximum 

modulus 27,730,000  pounds 

Vignole's  French  rails,  average  modulus. .  26,400,000       *' 
Dowlais   rails,   double  headed  minimum 

modulus 21,100,000       " 

*  Consulting  Engineer  of  the  Cincinnati  Southern  Ry. 

t  November  1st,  1875. 

+  These  experiments,  which  conclusively  prove  the  supe- 
riority of  the  American  system  of  bridge  details,  are  very 
complete,  and  will,  doubtless,  attract  much  attention.  There 
were 

below  20  millions  pounds 1  modulus 

from  20  to  25    "  "       7  moduli 

"    25  to  30    "  "       15   

"    30  to  34.6  "  "       5   


the  greatest  difference  being  31  per  cent. 
Morin  believed  that  the  great  variations  of 
moduli  (even  of  rails  of  same  section  and 
make)  should  be  explained  by  the  quality  of 
the  iron,  and  he  judges  that  the  better  metal 
should  show  the  higher  modulus.  But  the 
great  variations  also  of  moduli  of  bars  of  un- 
doubtedly excellent  make  and  of  great  uni- 
formity seem  to  disprove  his  judgment.  He 
states  that  he  has  met  with  moduli,  as  low  as 
17,000,000  pounds,  while  the  author  has  ob- 
served 18,000,000  as  a  minimum.* 


*  General  Morin  acknowledges  that  it  is  pretty  difficult  to 
determine  with  exactness  the  average  value  of  moduli  to 
suit  the  results  of  old  and  new  experiments,  and  he  says: 
But,  moreover,  it  must  not  be  lost  sight  of  that  it  happens 
pretty  often  that  iron  bars  of  the  same  manufacture,  fur- 
nished by  the  same  works,  present  notable  differences  in 
their  resistance  to  flexure. 

The  distinguished  French  officer  proposes  a  classification 
of  iron  of  high  grade  (average  modulus  of  30,000,000  pounds), 
ordinary  iron  (modulus  25,000,000  pounds),  and  soft  ductile 
iron  (modulus  from  21  to  even  17,000,000  pounds). 

But  this  classification  can  hardly  be  upheld,  since  the  very 
best  irons  (for  instance,  Swedes,  Russian,  &c.,  brands)  also 
are  the  softest  and  most  ductile  (and  as  regards  ultimate 
strength  somewhat  weak)  irons  which,  according  to  the 
classification,  would  have  to  belong  to  the  highest  and  also 
to  the  lowest  class.  The  experiments  on  eyebars,  now  parts 
of  existing  bridges,  as  made  by  the  author,  the  iron  being 
double  refined  (best-best)  have  given  moduli  from  the  low- 
est to  the  highest  class. 


There  seems  to  exist  this  law — that  the 
moduli  of  bars  of  same  section  made  from 
double  refined  iron  bars  (rolled  three  times, 
packeted  and  welded  twice),  such  as  called 
best-best,  are  more  uniform  than  bars  made 
from  best  iron,  such  as  were  used  by  Herr 
Malberg  in  the  Muhlheim  suspension  bridge. 
At  least,  many  thousands  of  bars  tested  at 
Phrenixville,  Pa.,  proved  to  be  remarkably 
uniform  in  their  moduli  as  long  as  they  were 
of  the  same  section,  whilst  the  moduli  were 
very  variable  when  bars  of  different  sections 
were  compared.  On  the  other  hand,  Styffe's 
experiments,  which  were  made  on  excellent 
steel  and  iron,  gave  a  maximum  tensile 
modulus  of  34,584,000,  and  a  minimum  of 
27,585,000  pounds,  which  is  similar  to  that 
of  an  iron  rail  from  Avon,  in  Wales. 

Morin's  experiments  on  flexure  of  unhard- 
ened  steel  gave  the  following  results : 

Maximum.    Minimum. 

From  Petin  &  Gaudet,  refined. . .  28,800,000    28,100,000 
"         "  "        puddled..  31,800,000    29,200,000 

"         "  u       crucible...  32,300,000    29,200,000 

Kvupp's 32,200,000    28,700,000 

"    mean  of  17  experiments..  30,300,000 

English 28,900,000 

It  should  be  noticed,  that  even  for  the 
finest  metal  that  we  know,  such  as  crucible 


40" 


steel,  the  variation  in  its  modulus  amounted 
to  11  per  cent.,  and  for  the  renowned  Krupp 
steel,  12  per  cent. 

The  most  accurate  experiments  are  still  to 
be  mentioned  (those  of  Herr  Woehler),  made 
by  suspending  the  test-piece  A  A,  so  that  it 

Fig.  7. 

I      * 


B 
(7    Si 


could  expand  freely ;  the  lever  L  L  carried 
the  load  P  and  the  arms  A  B,  A  B  were  ex- 
actly equal;  the  piece  B  J?,  acted  upon  by  a 
constant  moment  of  flexure,  carried  the  meas- 
uring apparatus  C  v,  v  C,  by  which  the  de- 
flection of  C  C  could  be  very  exactly  found. 
In  this  apparatus  the  deflections  of  the  test- 
pieces  become  comparatively  very  large,  and 
piece  C  C  was  free  from  the  influence  of 
knife  edges. 

These  are  Herr  Woehler's  results : 

Modulus  of  iron  from  Laura  huette 28,930,000  pounds. 

"  "  Phoenix  huette....  29,360,000       " 

"  "  Minerva  huette.  ..  31,680,000 

"  Low  Moor  iron 31,230,000       " 


41 


Modulus  of  "homogeneous  iron  from  Pear- 
son, Coleman    &  Co 32,340,000  pounds. 

Bochum  steel 32,000,000       ** 

Kmpp  steel ...31,600,000       *4 

All  these  materials  were  of  unusually  ex- 
cellent quality,  and  the  maximum  difference 
still  was  12  per  cent.* 

The  same  variability  which  we  have  found 
to  exist  between  iron  and  steel,  not  so  much 
as  to  the  quantity  of  carbon  contained,  as 
from  imperfections  of  manufacture  and  other 
causes  unknown  to  us,  was  noticed  with  the 
shearing  or  torsional  moduli.  Thus  Duleau 
found  for  iron  from  Perigord,  moduli  of 
14,450,000,  and  again  of  only  7,980,000 
pounds.  Iron  from  Arrieges  gave  8,450,000, 
English  Iron  9,860,000,  and  also  12,800,000 

*  Redlenbacher  quotes  the  moduli  of  iron  from  21,300,000 
to  35,500,000  pounds,  of  steel  from  28,500,000  to  34,100,000  Ibs 

Reuleaux  (Der  constructeur)  for  wire  bars  and  ordinary 
steel  gives  28,500,000,  for  cast  steel  (crucible  steel)  gives 
42,700,000. 

Kupffer,  in  St.  Petersburg,  by  experiments  on  sound  and 
flexure  of  small  specimens,  gets  from  25,000,000  to  30,000,000 
pounds. 

Coulomb,  Tredgold,  Lagerhjielm  and  Woshler  found  the 
modulus  of  hardened  steel  exactly  equal  to  that  of  unhar- 
dened  steel.  Kupffer  in  some  instances  finds  the  modulus 
of  hardened  steel  6)£  p.  c.  higher. 

Styffe  finds  that  the  modulus  of  cold  worked  iron  is  low, 
but  can  be  raised  by  exposure  to  a  glowing  heat.  He  also 
says  that  phosphorous  lowers  the  modulus. 


42 


pounds.  Wiebe  in  Berlin  quotes  the  shear- 
ing moduli  thus : 

Soft'  wrought  iron  9,000,000  pounds. 

Bariron 10,250,000       " 

Steel 9,000,000       " 

Finest  cast  steel  14,000,000       " 

These  figures  prove  that  we  cannot  know 
the  shearing  modulus  of  any  class  of  steel  or 
iron  without  direct  special  experiment. 

Our  conclusions  with  reference  to  the  sup- 
position of  a  constant  modulus  of  elasticity 
for  the  calculation  of  deflections  and  of  con- 
tinuous girders  are — from  known  and  un- 
known causes  : — First, .plain  iron  and  steel 
bars  vary  in  their  moduli  very  considerably. 
The  smallest  modulus  of  iron  was  found  to 
be  17,000,000,  the  maximum  above  40,000, 
000  pounds.  Single  refined  bars  of  same 
stock,  manufacture  and  section  vary  in  their 
moduli  by  35  per  cent.  Double  refined  (best- 
best)  bars  vary  little,  as  long  as  bars  of  same 
section  are  tested,  but  considerably  with  the 
sections,  the  minimum  being  18,000,000,  and 
the  maximum  above  40,000,000  pounds. 
The  moduli  of  rails  vary  by  30  per  cent.,  and 
similar  results  must  be  expected  from  com- 
mon angles,  beams,  channels,  &c.  Second, 
consequently,  riveted  bridge  members  com- 


43 


posed  of  angles  and  plates  of  various  thick- 
ness and  manufacture,  interrupted  in  their 
homogeneousness  by  punched  holes,  covering, 
reinforcing  splice-plates,  &c.,  must  neces- 
sarily show  still  greater  variations  in  their 
moduli  than  was  found  for  plain  integer  bars. 
Third,  the  hypothesis  of  a  constant  modulus 
of  elasticity  of  the  material  of  a  bridge  being 
unfounded,  the  theory  built  on  such  hypo- 
thesis should  be  abandoned.* 

Having  arrived  at  such  conclusion,  we 
nevertheless  must  expect  to  hear  an  objection 
against  its  logical  consequences,  namely  this 
— that  numbers  of  continuous  bridges  do 
good  service  in  practice.  So  they  hitherto 
have  done,  not  because  the  principle  of  conti- 
nuity is  admissible,  but  because  the  factor  of 
safety  used  in  their  construction  has  hidden 
the  error  made  in  their  design.  For  the  same 
reason,  the  Victoria  bridge  in  Canada  stands, 
which  is  made  continuous,  but  simply  by 

*  Mr.  Baker  most  pertinently  remarks  with  reference  to 
continuous  girders :  "  The  most  expert  mathematician 
would  have  to  devote  a  month  or  more  to  the  preliminary 
calculations  of  a  very  ordinary  bridge,  and  the  result  de- 
duced would  not  after  all  be  more  reliable  in  practice  than 
those  arrived  at  by  comparatively  simple  modes  of  investi- 
gation, chiefly  on  account  of  the  varying  elasticity  of  different 
portions  of  even  the  same  plate  of  iron." 


44 


combining  two  single  spans  whose  greatest 
chord-sections  are  in  their  centres,  whilst  the 
greatest  chord-strains,  according  to  theory, 
would  fall  where  the  cross-sections  are  made 
the  smallest.  For  the  same  reason,  conti- 
nuous draw  bridges  stand,  which  we  find 
composed  of  two  halves,  each  designed  as  a 
single  span.  The  author  knows  of  one  instance, 
that  a  Hodgkinson  cast  iron  beam  was  put  in 
place  upside  down,  so  that  the  heavy  tensional 
flange  was  under  compression  while  the  com- 
pressional  flange  of  only  one-fifth  the  area  of 
the  tensional  one  was  strained  under  tension, 
and  yet,  on  account  of 'the  factor  of  safety, 
the  beam  stood. 

III. — OTHER  DEFICIENCIES  OF  CONTINUOUS 
GIRDERS,  as  regarding  the  imperfections  of 
the  theory,  the  danger  from  defective  manu- 
facture, from  settling  of  piers,  and  the  in- 
crease of  strains  by  the  action  of  the  heat  of 
the  sun  and  the  omission  of  the  influence  on 
the  strains  caused  by  deflections  due  to  the 
web  systems. 

In  order  to  find  the  exact  extension  or  com- 
pression of  a  member  of  a  bridge,  we  must 
know  not  only  the  modulus  and  the  total 
strain  of  the  member,  but  also  its  cross-sec- 


45 


tion.  The  problem  of  continuous  girders, 
however,  is  to  find  this  very  section.  The 
theory  assumes  that  all  sections  are  equal,  or 
at  least  that  the  moment  of  inertia  of  a  girder 
or  a  bridge  is  a  constant  throughout.  Under 
this  supposition  we  get  smaller  chord  strains 
over  the  middle  piers  than  exist  in  reality. 

In  the  case  of  two  equal  continuous  spans 
under  full  load,  with  uniform  moment  of 
inertia,  we  find  the  moment  of  flexure  over  a 
middle  pier  to  be  equal  0.125  I2  p,  where  p 
represents  the  total  load  per  lineal  foot,  and 
I  denotes  the  length  of  each  span  in  feet. 
But  if  we  suppose  that  the  same  bridge, 
under  full  load,  shall  be  strained  equally  per 
square  inch,  the  co-efficient  0.125  becomes 
0.1464,  which  indicates  strains  over  the  mid- 
dle piers  15  per  cent,  higher.  In  reality,  the 
continuous  bridge  being  not  perfectly  varied 
in  chord  sections,  the  difference  will  be  less  ; 
but  it  may  be  remarked  that  the  chords  of  a 
continuous  bridge,  properly  designed  accord- 
ing to  specification,  would  only  be  about  10 
per  cent,  lighter  than  those  of  equal  single 
spans.  With  an  enormous  amount  of  labor, 
this  deficiency  of  the  ordinary  theory  can  be 
corrected,  and  it  has  been  done  in  a .  few 


46 


bridges.  But  considering  the  irregularity  of 
the  moduli,  such  labor  seems  superfluous. 

A  serious  cause  of  errors  in  the  construc- 
tion of  continuous  girders  refers  to  the  distri- 
bution of  strains  over  the  posts  and  ties  in 
case  that  two  or  more  web  systems  have  been 
adopted.  In  a  single  span  bridge,  a  load 
brought  on  a  panel  joint  of  one  separated 
web  system,  being  split  into  two  shearing 
forces  in  accordance  with  the  law  of  the 
lever,  there  cannot  be  any  mistake  about  the 
strain  in  a  web  member,  as  long  as  the  end 
posts  are  vertical,  and  if  they  are  inclined, 
the  error  can  amount  to  only  one  increment 
of  one  panel  load. 

The  problem  of  web  strains  with  conti- 
nuous girders  depends  not  only  on  the  law  of 
the  lever,  but  also  on  the  angles  of  deflection 
a,  ft,  y,  6 — not  only  of  one,  but  of  all  spans 
together.  We  remember  that  by  the  moments 
M^  MV  MV  &c.,  forces  +p^  ±p»  <&c.,  were 
originated,  which  disturb  the  law  of  the 
lever.  If,  therefore,  in  a  continuous  bridge 
there  are  two  or  more  web  systems,  we  are 
utterly  ignorant  as  to  the  distributipn  of  the 
reactions  over  the  two  or  more  systems  which, 
at  every  end  pier  and  at  every  middle  pier 


are  connected.  How  much  of  piy  p^  p^  <fcc., 
is  acting  in  one,  and  how  much  into  the  other 
system  ?  This  we  do  not  and  cannot  know, 
for  the  distribution  of  the  reactions  will  de- 
pend entirely  on  variations  due  to  manufac- 
ture, in  the  mill  as  well  as  in  the  shops.*  It 
may  even  happen  that  a  member  of  one  sys- 
tem receives  tension  and  the  other  compres- 
sion. It  is  therefore  very  desirable  that  ocn- 
tinuous  girders  should  be  built  with  but  one 
web  system. 

Hitherto  in  all  our  investigations  we  have 
made  the  supposition  that  the  erection  of  con- 
tinuous girders  was  of  such  perfection  that 
the  single  spans  were  connected  under  the 
action  of  moments  Ml  M'^  &c.9  which  accord- 
ed completely  with  a  perfect  theory.  Even 
with  the  best  staging  and  under  the  supposi- 
tion of  the  most  careful  workmanship  it  will 
be  hard  to  perfectly  fulfill  this  condition.  But 
suppose  it  were  possible,  and  that  a  pier  set- 


*  In  other  words :  Two  or  more  systems  of  braces  and 
fties  in  webs  of  single  span  bridges  can  be  made  perfectly 
independent  of  each  other,  and  the  strains  in  each,  there- 
ore,  can  be  calculated  by  the  law  of  the  lever  perfectly 
independent  of  each  other,  whereas  in  each  continuous 
bridge  with  more  than  one  web  system  these  systems  are 
connected  together  over  the  piers,  therefore  never  are  inde- 
pendent of  each  other,  and  can  not  be  calculated  separately. 


48 


tied.  In  this  instance,  the  girder  would  re- 
ceive considerable  disturbances  of  its  strains, 
which  in  some  points  would  be  decreased, 
while  in  others  increased.  The  deeper  the 
girder,  the  greater  the  disturbance  from  this 
cause  would  become,  so  that  it  seems  advis- 
able to  leave  to  the  girder  as  much  plasticity 
as  possible,  by  adopting  a  depth  smaller  than 
demanded  by  simple  theoretical  economy.  In 
fact,  this  change  in  the  value  of  calculated 
strains  could  become  enormous  ;  hence  the 
piers  of  continuous  girders  should  be  built 
more  substantially  than  is  necessary  for  single 
spans.  But  this  caution  is  costly.* 

*  It  happens  not  unfrequently,  that  settling  of  piers  of 
draw  bridges  causes  difficulties  in  turning  the  superstruc- 
ture. A  bridge  of  this  kind  near  New  Haven,  Connecticut, 
(Quinnipiat  Bridge),  was  commenced  two  years  ago,  but  is 
not  yet  in  operation.  The  central  pier  tipped,  the  super- 
structure had  to  be  jacked  up,  the  masonry  to  be  partly  re- 
moved and  newly  laid.  The  calculations  for  the  superstruc- 
ture of  this  bridge  had  been  made  with  great  painstaking, 
involving  much  algebraic  labor,  which  thus  was  most  essen- 
tially vitiated  by  the  nature  of  the  substructure.  Similar 
instances  have  happened  elsewhere  and  where  noticed,  be- 
cause the  turning  gear  readily  indicated  the  disturbance 
below.  Of  single  span  bridges  being  in  their  strains  inde- 
pendent from  the  heights  of  support,  we  rarely  hear  of  com- 
plaints caused  by  the  settling  of  the  piers. 

The  cost  of  bridge  foundations  and  masonry  differs  between 
wide  limits,  according  to  quality,  and  for  continuous  bridges 
the  very  best  class  of  either  would  be  required. 


49 


The  real  economy  of  continuous  girders,  as 
claimed  in  Europe,  when  compared  with  sin- 
gle span  lattice  bridges,  consisted  in  building 
the  girders  on  land  and  then  rolling  them  over 
the  piers.  This  mode  of  erection  is  elegant, 
but  it  does  not  fully  secure  the  lit  of  the  super- 
structure to  its  bearings  on  the  piers,  and  it  is 
still  doubtful  whether  this  mode  of  erection 
always  can  compete  with  that  of  single  spans, 
designed  with  the  specific  American  details,  f 
In  the  subsequent  example  of  two  200  feet 

T  iu  ihe  main  building  of  the  .Philadelphia  Exhibition  the 
North  Eastern  R.  R.  of  Switzerland  has  laid  out  a  report  on 
their  bridges. 

This  official  report  is  interesting  in  many  ways.  One  re- 
mark thereof  is ;  "  The  erection  of  ironwork  on  scaffolds  is 
preferred  to  the  method  of  pushing  the  girders  over  the  piers. 
This  latter  method  never  is  allowed  without  intermediate  tem- 
porary supports,  and  without  reinforcement  (Armirung)  of 
the  girders." 

The  acknowledged  best  builder  in  Switzerland  always  uses 
false  works,  but  the  mentioned  French  works  push  their 
continuous  girders  over  the  piers.  However,  it  was  observed 
that  they  thereby  were  likely  to  furnish  second  class  work, 
and  it  happened  that  the  violence,  or  the  undue  strains,  con- 
nected with  this  method,  caused  rivet  heads  to  fall  oft'. 

In  case  the  method  of  pushing  continuous  girders  over  the 
piers  is  not  to  be  used,  their  erection  becomes  more  expen- 
sive than  that  of  single  spans.  Not  only  that  these  scaffolds 
must  be  very  unyielding  and  substantial,  but  all  spans  of  the 
same  set  of  continuous  girders  must  be  provided  with  false 
works  at  the  same  time,  whereas,  in  the  erection  of  single 
spans,  only  a  scaffold  for  one  span  is  used,  or  is  usually  used 
repeatedly.  Also  the  risk,  by  erecting  two  or  three  spans  at 


50 


continuous  girders,  we  shall  give  figures  as  to 
the  disturbance  of  strains  in  case  the  girders 
do  not  fit  their  supports.J 

What  has  been  said  as  to  the  disturbances 
of  strains  by  settlement  of  piers  or  by  badly 
executed  girders,  is  equally  true  in  regard  to 
the  effect  of  the  sun.  Swing  bridges  have 
been  drawn  crooked  by  the  rays  of  the  sun 
falling  upon  one  side.  In  others,  the  bottom 
chords  are  covered  by  floor  timbers,  and  the 
top  chords  are  considerably  overheated  by  the 
sun,  or  unequally  cooled  under  sharp  winds. 
The  effect  of  this  unequal  temperature  is  en- 
ormous, and  it  is  sufficient  (even)  to  raise  a 
continuous  bridge  from  a  pier.  In  the  case 
of  tubular  girders,  this  objection  has  peculiar 
force.  Hereafter  we  shall  take  examples 
and  calculate  the  strains  caused  by  change  of 

the  same  time,  is  considerably  increased.  The  economical 
method  of  pushing  girders  over  piers  in  a  few  rare  instances, 
and  under  proper  precautions  (it  was  proposed  for  the  Ken- 
tucky River  Bridge,  Cin.  S.  R.  R.),  might  be  used  when 
the  girders  finally  could  be  separated  again  by  establishing 
hinges  in  alternate  spans. 

}  In  1867  and  later,  the  writer  gave  attention  to  the  practi- 
cal solution  of  the  old  idea  of  weighing  the  reactions  of  con- 
tinuous girders,  by  means  of  hydraulic  presses,  and  designed 
a  cheap  apparatus  to  accomplish  this  purpose.  But  the  plan 
had  previously  been  tried  in  the  erection  of  a  bridge  in 
Silesia,  Prussia. 


51 


temperature  of  the  top  and  bottom  chords  of 
continuous  girders. 

The  last  objection  urged  against  continuous 
girders,  refers  to  the  mode  of  proportioning 
those  parts  of  their  chords  which  at  each 
passage  of  a  train  have  to  stand  pressure  as 
well  as  tension.  The  space  for  two  spans 
thus  strained  equals  33  per  cent,  of  the  length 
of  each  chord.  The  European  practice  to  pro- 
portion these  parts  is  to  find  the  maximum 
total  strain,  divide  it  by  the  maximum  speci- 
fied strain  per  square  inch,  and  make  the 
actual  section  as  near  to  this  theoretical  sec- 
tion as  can  be  done.  This  is  radically  wrong. 
Herr  Woehler's  experiments,  perhaps  the  most 
thorough  ever  made,  extending  over  a  time  of 
more  than  12  years,  have  established  beyond 
doubt,  that  the  strain  which  controls  the  dur 
ability,  equals  the  sum  of  the  maximum  ten. 
sion  and  compression  of  a  chord  piece.  A  bar 
strained  tensily  to  35,200  pounds  per  square 
inch,  can  stand,  say,  100,000,000  repetitions 
of  such  strains,  but  if  at  the  same  time  strain- 
ed compressively  to  35,000  pounds,  it  will 
break  after  a  small  number  of  repetitions,  say 
100,000,  whereas  if  strained  to  the  limits  of 
4-  17,600  pounds,  it  will  show  as  much  dur- 


52 


ability    as    if    tensively   strained    to    35,200 
pounds.* 

Writers  on  continuous  girders,  generally 
erred  in  comparing  girders  of  various  systems 
of  the  same  depth,  whereas  the  proper  depth 
of  girders  is  a  measure  peculiar  to  each  sys- 
tem of  design  and  essentially  depending  on 
the  relative  quantities  of  chord  and  web- 
strains.  The  smaller  the  web-strains,  the 
deeper  a  girder  can  be  built.  But  the  web 
material  needed  for  continuous  girders  ex- 
ceeds that  of  single-span  girders  by,  say,  10 
per  cent.,  while  the  material  necessary  for  fohe 
chords  (according  to  theory)  is  just  about  as 
much  less.  The  consequence  is,  that  an  in- 
crease of  depth  increases  the  web  material 
more  rapidly  than  is  the  case  for  single-span 
girders.  Because  of  this,  probably,  parabolic 
girders  were  built  much  deeper  in  Europe 
than  quadrangular  trusses,  and  there  is  no 
reason  why  this  same  principle  should  be  ap- 

*  For  further  information  on  this  subject,  compare  Herr 
Woehler's  Report  in  the  Berliner  Zeitschrift  fur  Bauivesen. 
These  experiments  were  continued  by  Professor  Spangen- 
berg,  of  which  a  translation  has  appeared,  published  by  Van 
Nostrand,  New  York.  See  also  Inspecting  Engineer  Mutter's 
article  in  the  Zeitschrift  des  Oestreichischen  Ingenieur  and 
Archftecten  Vereius,  1873.  See  also  Enbkam's  Zeitschrift* 
1875. 


plied  to  continuous  girders.  Even  an  engin- 
eer like  Prof.  Kulmann  in  Zurich,  made  the 
mistake  of  comparing  parabolic,  continuous, 
quadrangules,  single  and  Warren  girders,  by 
supposing  all  of  them  to  be  of  the  same 
depth,  namely,  one  tenth  of  their  lengths. 

For  comparison,  we  here  give  a  few  figures 
taken  from  the  calculations  for  the  new  Buda- 
Pest  bridge  f  in  Hungary,  now  under  progress 
of  erection.  There  will  be  4  spans,  of  321 
feet,  carrying  two  tracks,  depth  32  feet,  each 
calculated  for  3,000  pounds  per  foot  ;  strains 
9,740  pounds  per  square  inch  ;  compression 
correspondingly  ;  weight  of  parabolic  trusses 
285,500,  and  of  continuous  lattice  trusses  270, 
300  kilogrammes  (two  spans  each).  We  know 
that  American  trusses  of  proper  proportions 
can  be  built  lighter  and  cheaper  than  para- 
bolic trusses,  and  therefore,  also,  in  this  in- 
stance there  is  no  reason  for  giving  on  the 
score  of  greater  economy  the  preference  to 
continuous  lattice  trusses.  But  the  contractor 
had  made  a  very  low  bid  and  desired  the  con- 
tinuous bridge  to  be  chosen,  though  origin- 
ally, single  spans  were  designed  and  bid  upon. 
These  continuous  girders  are  intended  to  be 

See  Stammer"  *  Enyineei\  Vienna,  1875. 


54 


rolled  over  the  piers.  The  depth  of  one-tenth 
is  decidedly  too  low  for  single  spans  ;  it 
should  have  been  taken  at  40  instead  of  32 
feet. 

Herr  Schwedler,  in  Berlin,  who  certainly 
has  as  much  experience  in  European  bridge 
building  as  any  other  engineer,  and  who  is 
so  much  an  authority  in  theoretic  matters 
that  not  even  the  most  distinguished  theorist 
can  very  well  set  him  aside,  in  1865*  had 
made  it  his  strict  rule  neither  to  build  nor  to 
recommend  continuous  girders  or  arches  with- 
out at  least  hinges  at  the  skewbacks.  He 
builds  a  species  of  bow-string  girders  with 
depths  of  one-seventh  of  the  span,  which, 
though  to  American  eyes  complicated  in  de- 
tails, yet  are  decidedly  superior  to  continuous 
girders.  In  England  far-going  mathematical 
deductions  on  this  topic  have  not  been  studied 
as  much  as  in  France,  and  in  imitation  of  the 
French  engineers  in  Germany.  Nor  has  the 
distinguished  late  Professor  Rankine  dwelt 
on  this  subject  very  extensively.  English 


*  See  Herr  Schwedler's  theses  on  bridge  building  in  the 
Zeitschrift  fur  Bauwesen.  The  very  excellent  scientific 
pocketbook,  Des  Ingenieurs  Taschenbuch  des  Verein  Hutte, 
10th  edition,  Berlin,  does  not  treat  continuous  bridges. 


55 


engineers  of  name  have  expressed  themselves 
that  the  expectations  of  those  continental 
engineers  were  higher  than  could  be  realized 
in  practice,  that  much  more  was  to  be  done 
in  advancing  practical  knowledge  by  means 
of  well  devised  and  well  conducted  experi- 
ments carefully,  logically  and  rigidly  inter- 
preted, than  by  the  application  of  hypotheses 
and  mathematical  reasonings,  many  of  which 
simply  concealed  real  ignorance,  that  there 
were  few  locations  in  which  continuous  gir- 
ders were  to  be  preferred,  and,  generally 
speaking,  the  circumstances  were  such  that 
there  would  be  no  saving  of  money  in  their 
use. 

Others  have  declared  that  the  question  of 
continuous  trusses  was  too  complicated  for 
investigation,  or  that  the  formulae  were  too 
troublesome  in  application,  <fcc.,  &e.  This 
last  objection,  however,  is  only  in  part  rele- 
vant. If  the  theory  as  given  in  text  books, 
simplified  as  it  now  is  by  the  theorem  of  the 
three  moments  were  based  on  sufficiently 
correct  hypothesis  and  were  collect  by  itself, 
theoretically  speaking,  there  would  be  no 
objection  to  the  application  in  case  of  large 
bridges.  But,  unfortunately,  the  theory  it- 


self,  if  applied  to  skeleton  trusses,  is  defi- 
cient. 

This  deficiency,  due  to  the  omission  of  the 
web  system,  is  purely  analytical,  and  yet  has 
escaped  the  notice  of  the  undoubtedly  very 
able  French  mathematicians  to  whom  we  owe 
the  development  of  this  branch.  The  correct 
introduction  of  the  consideration  of  the  web 
system,  very  variable  as  it  is,  in  the  unkitoim 
sections  of  its  members,  would  be  a  sheer 
ttnfthematical  impossibility. 

The  first  application  of  the  theory  to  iron 
structures  probably  was  made  in  England. 
There  were  to  be  built  shallow  plate  girders, 
whose  web  plates  being  of  constant,  or  nearly 
constant  thickness,  participated  in  the  resist- 
ance to  the  moments,  and  whose  influence  on 
the  deflections  was  very  small  indeed.  Here, 
then,  one  important  objection  to  the  common 
theory  did  not  exist. 

Moreover,  there  actually  was  a  real  eco- 
nomy in  connection  with  the  principle  of  con- 
tinuity of  plate-girders.  For  the  web  plates 
of  these  girders  practically  could  not  be  re- 
duced in  thickness  to  the  thereotical  require- 
ments, and  web  plates  such  as  practically 
could  be  used  for  single  span  bridges  were 


also  strong  enough  to  bear  the  increased  web 
strains  of  continuous  girders.  Moreover,  the 
expense  of  such  webs  for  large  spans  com- 
pelled to  small  depths.  And  since  no  mate- 
rial could  be  saved  in  the  plate-webs,  the 
theory  of  continuity  offered  a  welcome  help 
toward  reduction  at  least  of  the  chord  mate- 
rial. 

The  modulus  of  elasticity  once  accepted  to 
be  a  constant  value,  it  was  entirely  rational 
and  economical  to  use  continuous  plate-gir- 
ders. On  the  contrary,  it  is  theoretically 
wrong  and  practically  not  economical  to 
build  continuous  skeleton  structures.  In  other 
words,  the  huge  English  continuous  plate - 
girders  and  their  French  imitations  are  more 
scientific  than  the  extension  of  the  ordinary 
theory  to  open- webbed  trusses. 

Strict  logic  led  many  French  engineers  to 
adopt  constant  sections  of  chords,  so  that 
ft/tit/  in  case  of  more  than  two  continuous 
spans  by  the  mode  of  proper  proportions  as  to 
their  lengths  real  economy  could  be  secured. 
And  it  also  must  he  remarked  that  it  like- 
wise was  a  fully  logical  reasoning  of  these 
engineers  to  seek  greater  equaliaztion  of  mo- 
ments by  lowering  the  bearings  on  the  middle 


58 


piers  of  continuous  girders  of  two  or  more 
than  two  spans.  Thereby  pressure  would  be 
produced  in  the  top  chords  over  the  middle 
piers,  so  that  the  great  negative  moments  at 
these  points  (tension  in  top  chords,  pressure 
in  bottom  chords)  would  be  reduced,  and  the 
smaller  positive  moments  between  the  piers 
would  be  increased.  This  construction,  theo- 
retically, is  rational,  if  the  chord  sections  are 
made  of  constant  sections  ;  but  it  becomes  at 
once  irrational  and  even  more  expensive  if 
the  chords  are  varied.  Also,  if  the  chords  are 
varied,  there  is  no  longer  any  reason  why 
the  spans  should  not  be  equally  long.  And, 
in  fact,  it  would  be  slightly  more  economical 
to  arrange  purposely  for  great  negative  mo- 
ments over  the  middle  piers. 

In  many  treatises  on  continuous  girders 
there  exists  great  confusion  as  to  alleged  ad- 
vantages of  lowering  of  middle  piers,  and  of 
best  proportions  of  end  and  middle  spans. 

We  have  now  explained  why  we  cannot  ad- 
mit that  it  is  desirable  that  American  bridge 
engineers  should  seriously  regard  continuous 
girders,  however  attractive  they  may  be  to 
some  mathematicians  on  account  of  the  wide 
field  for  interesting  problems  presented,  and 


,59 


we  shall  proceed  to  briefly  lay  down  the  theory 
such  as  derived  from  the  suppositions  which 
were  found  questionable,  whereupon  an  ap- 
plication shall  be  made  to  two  200  feet  rail- 
way spans  in  comparison  with  single  spans. 
There  we  shall  find  occasion  to  test  all  what 
has  been  said  in  previous  paragraphs,  also  to 
examine  the  probable  or  possible  errors  of 
design,  and  thus  to  arriv§  at  final  conclu- 
sions. 

This  investigation  will  have  a  negative 
and  also  a  positive  value  ;  negative,  because 
we  spend  useful  time  on  the  study  of  an 
objectionable  system,  and  positive,  because 
we  once  more  will  have  occasion  to  learn 
that  in  our  art  also,  the  most  perfect  system 
must  be  the  most  simple  one. 

IV. — GENERAL  DEVELOPMENT  OF  A  SIMPLE 
METHOD  OF  FINDING  THE  PRINCIPAL  FORMULA 
OF  CONTINUOUS  GIRDERS. — The  angles  a,  ft, 
6%  y,  <fcc.,  can  be  found  by  considering  one 
single  formula  developed  in  the  theory  of 
single  span  beams,  which  in  the  following 
paragraph  will  be  used  repeatedly.  For  the 
present,  we  make  the  same  suppositions  which 
are  used  by  other  writers,  though  contested 
by  us.  We  assume  therefore,  first.  The 


60 


modulus  of  elasticity  of  iron  is  a  constant 
and  known  value.  Second.  The  continuous 
girders  throughout  have  the  same  cross-sec- 
tion, moreover  they  have  straight  parallel 
chords.  Third.  The  deflections  of  these  gir- 
ders are  not  modified  by  the  shearing  forces  ; 
in  other  words,  the  struts  and  ties  of  the  wel> 
do  not  change  their  lengths.  Fourth.  The 
web  systems  of  the  girders  are  so  arranged 
that  there  is  no  doubt  of  the  office  of  each 
separate  system  of  struts  and  ties,  which  con- 
dition can  only  be  fulfilled  in  case  of  but  one 
system  of  diagonals  and  posts.  Fifth.  The 
temperature  of  all  members  is  alike,  and  can- 
not change  in  any  separate  member.  We  use 
this  notation  :  E  is  the  modulus  of  elasticity 
in  pounds  per  square  inch,  which,  as  known, 
is  the  measure  of  stiffness  of  material,  the 
greater  the  modulus  or  the  less  the  value  of 

-j-j  the  less  proportionally  are  the  elastic  de- 
formations. I  is  the  moment  of  inertia  of 
the  girder,  equal  for  a  skeleton  truss,  to  the 
cross-section  of  one  chord  multiplied  by  one 
half  the  square  of  depth  of  the  truss,  all 
dimensions  taken  in  inches.  Like  E,  the 
value  I  stands  in  inverse  geometrical  propor- 


61 


tion  to  the  deflection  of  a  beam  or  girder. 
Pv  Pz  •  •  •  Pn+i  denote  the  elastic  reactions 
in  pounds  caused  by  the  unknown  moments 


N 
<8i?                 / 

y^- 

rfW^ 

)//j///cyFM(nwrz 

\ 

\ 

§ 

'    k 

11 

'ft* 

1 

II 
I 

/ 

1 

11 

/ 

<^^ 

/ 

-4f  — 

--O  -  - 

"/<?' 

I 

j 

•^  i 

,    •*//- 

I 

i*" 

^JUdUMJV 

f 

$ 

i 

/ 

1 

i 

1 

1 

/I 

N, 

/^ 

/N) 

l*i 

f 

1 1 


y:i 


62 


Ml  M2  M3  .  .  .M>1{  over  the  middle  piers 
of  continuous  girders.  ^  /2  /4  .  .  .  in  are 
the  lengths  of  the  spans  in  inches,  conse- 
quently Ml  MZ  -  •  •  -^fn-i  must  be  measured 
in  pound  inches. 

The  above  figure  represents  a  truss  A  B, 
which  is  supposed  to  be  acted  upon  by  no  other 
forces  but  the  pair  +  ^  —  S19  which  create 
a  moment  M=  Sh  counteracted  by  a  force 
p  in  A.  This  force  p  in  combination  with 
—  p  in  B  on  the  lever  /  (=  A  B)  has  the  tend- 
ency to  turn  the  truss  A  B  in  opposite  direc- 
tion to  My  and  to  produce  equilibrium;  con- 
sequently pi  must  equal  M.  The  sum  of  the 
horizontal  as  well  as  of  the  vertical  forces 
being  zero,  no  movement  of  the  truss  A  13 
will  be  possible;  nevertheless  its  elasticity 
will  cause  a  flexure  which  increases  in  curva- 
ture from  A  to  B.  This  is  due  to  the  mo- 
ments of  flexure  increasing  in  geometrical 
progression  from  A  to  B,  which  moments  in 
the  triangle  A  B  C  are  represented  by  the 
straight  line  A  C.  The  maximum  moment 
occurs  at  B  and  is  •=.  M^=.  Sh"=.pl.  For 
any  distance  x,  from  A  the  moment  will  be 
MX  •=.  px. 

The  above  figure  also  represents  that  the 


63 


tx)  tal  strains  in  the  chords  increase  in  geo- 
metrical proportion  from  A  to  B.  At  B  the 
total  strains  will  be  S  and  —  8,  in  A  the 
strains  will  be  zero.  The  chords  being  sup- 
posed to  be  equally  strong  in  section,  the 
strains  per  square  inch  likewise  increase  in  a 
geometrical  progression  from  A  to  B.  The 
web  strains,  however,  remain  constant  through 
the  whole  girder,  because,  according  to  the 
nature  of  this  problem,  the  shearing  force  has 
a  constant  value  ==/>.* 

We  know  that  the  expressions  for  the  an- 
gles a  and  y6>  must  contain  E  and  I  as  divi- 
sors, and  I  and  the  maximum  moment  as 
multipliers,  so  that  we  only  need  find  the  co- 
efficient to  this  expression.  Actually  the 
development  gives: 


__ 
~  6  E  T~6  E  l) 
and  }  (III.) 


so  that  ft  is  twice  as  great  as  <r  ;  )see  Fig.  8). 

•In  case  the  truss  A  .B  had  been  perfectly 

varied  in  sections  to  suit  the  moments,  the 

*  In  the  sequel  it  will  be  shown  how  the  formulae  for  the 
angles  a  and  J5,  can,  under  the  suppositions  made,  be  found 
without  the  aid  of  the  inflnitessimal  calculus. 


64 


coefficient  of  /3  would  no  more  be  i  but 
would  have  increased  to  i,  which  is  50  per 
cent,  more  than  under  the  supposition  of  a 
constant  moment  of  inertia  I,  for  any  section 
of  the  truss  A  Bj  which  result  indicates  need 
of  caution  in  making  this  supposition  for  con- 
tinuous girders. 

The  simple  law  contained  in  Eq.  (Til)  is 
sufficient  to  easily  solve  the  remainder  of 
questions  embodied  in  the  theory  of  continu- 
ity. Suppose  the  girder  A  B  to  be  acted 
upon  by  this  moment  Ml  in  A  and  by  the 
moment  J/,  in  B,  both  moments  acting  to- 
wards an  increase  of  upward  flexure.  What 
will  be  the  angles  a  and  ft?  This  problem 
is  only  a  corollary  to  the  first.  We  have: 
end  points  A,  B\  moments  at  these,  Mv  M^: 

M,  I  ,,     Ml  I 

Angles  due  to  M.  at  A  =  -T-  ~ 

o 


2  6  E  I  3  El 

Totalansles  ..................  *= 


~  6  E  1^  3  El 

In  case  Ml  were  =   M^  there  would  be 
throughout   the   girder   a   constant   moment 


65 


of  flexure,  and   a  would   become   equal   to 

M I 

ft  •=.  -— _,-  T .     In  this  instance,  the  elastic  line 

would  be  uniformly  curved,  and  part  of  a 
circle  whose  radius  is  p  —  -^ ,  as  well 

known  from  the  theory  of  single  spans. 

Finally,  we  have  to  determine  the  angles 
y  and  fi  of  a  single  span  exerted  by  a  single 
panel  load  P.  In  Fig.  9,  P  represents  the 
panel  load  at  the  distances  a  and  b  from 
points  A  and  7?,  a  +  b  being  equal  to  A  B  •=.  I. 
By  the  law  of  the  lever,  the  reaction  of  the 

pier  at  A  will  be — — and  at  pier.Z?,  it  will  be 
l/ 

P  a 

—j—.     •&  &  representing  the  tangent  on  the 

elastic  curve  at  Z>,  the  angles  ft  and  y#1?  are 
known  as  well  as  the  angles  a  and  orr  The 
angles  q>  -f-  ?/?  together  must  equal  ft  +  ft^. 
(Consider  that  (p  +  if>  +  angle  D  =  180°, 
and  that  0  +  /?t  +  D  also  =  180°).  The 
angles  of  deflection  being  very  small,  can  be 
considered  as  equal  to  their  tangents,  namely : 

d  d 

(p  --  — ;  tp  ---T  /  and  cp  :  ip  -—  b  :  a.     But  on 

Ma  Mb 

the  other  hand  ft  nr-and  ftl  =    -         so 


67 


that  /3l  :  fi=b  :  a9  and  sine?  <p  4-  */> —  > 
apparently  /?t        ^?  and  ft       ?/?,  so  that  sim- 

p!y: 

__  ff  M  a     .     M  6 


M  6       Jlf  a         Pab 

(2 


In  case  the  girder  A  13  should  have  carried 
any  number  of  loads,  Pl  _P2  &c.,  with  distan- 
ces «!  ^>1?  «8  £2,  «3  53,  &c.,  there  would  have 
been  — 

1       v  f  I  ^e  sum  °^  a^  i 
~~  6  1  E~J       II     expressions    ) 

,      2  Pa  b  (a  +  2  b)} 
)=  6/A7 

1  f  (  the  sum  of  all  | 

~~  6lTT~i  ^11    expressions    ) 


Now  we  are  prepared  to  write  the  final 
formula  of  continuous  girders.  The  equil- 
ibrum  of  the  moments  M19  Ma  and  M8  with 
the  forces  P19  P2,  &c.,  and  Q19  QZ9  §3,  is 
found  from  (Eq.  /.)  #2  +  y9  =n  //2  -|-  a3  where 


68 


/, X -12~-  -    ?3 

$2  -f-  ys  are  the  angles  of  deflection  due  to 

angles  of  elevation  due  to  M1?  M,  and  M3 — 
of  the  spans  considered  as  single  ones. 

Their  values  are  :  tf,  = — -==-=— 2  [Pa  b 
(2  a  -f-  b)~\  for  span  /2, 


7s  =-Q-JzTj'2  [  6  «  ^  (2 


«)]  for  span  /8. 


__ 

P*  ~~6  E  I~ 
consequently 
6  J7  J  (tf,  + 


.        _. 
E  I>     3~6 


(I, 


( 


which  actually  is  the  equation  of  Henry  Ber- 
tot  ;  also  : 

~  2  [Pa  b(a  +  2b)+~-  2  [Qab  (2  a 

+  ft)]  =  M,  I,  +  2  Jff  (/,  +3/3)  +  M,  13(  VI.) 

This  equation  is  of  the  first  degree  and  con- 

tains three  unknown  quantities,  viz.:  M19  Ma, 


69 


M8,  whilst  the  expression  on  the  left  side  is 
fully  known  since  the  loads  JP19  P^  P^  <fcc., 
and  Q19  Q^  $3,  with  their  distances  a  and  b 
from  the  end  points  of  each  truss  are  given 
quantities. 

At  every  central  pier  of  a  continuous  girder 
there  is  an  equation  of  this  form,  and  there  is 
also  an  unknown  moment,  so  that  we  have  as 
many  equations  of  the  first  degree  as  there 
are  unknown  moments.  The  problem  to  find 
these  moments  consequently  is  solved  analyti- 
cally, though  the  labor  of  solving  these  equa-- 
tions  algebraically,  in  case  of  many  spans,  is 
rather  tedious. 

If  we  introduce  for  a  and  b  the  correspond- 
ing number  of  panels,  call  I  =  n  d,  where  d 
is  the  length  of  each  panel,  put  a  =  m  d  and 
b  =  / — a  =  (n — m)  d,  we  arrive  at  these  sim- 
plifications for  the  expressions  on  the  left  side 

of  Eq.  ( VI); 

instead  of       2  [Pa  b  (a -f-  2  #)],  we  get 
I/ 

-  2  [Pm  (n  —  m)(2n  —  m)]  (A.) 
TL 

instead  of  —  2  [Pa  b  (2  a  -f-  #)],  we  get 
t 

*  2  [Pm  (n*  —  m*)]  (3.) 


The  expressions  A  and  B  for  the  spans  lt  /2 
13  /4,  &c.,  4  being  denoted  by  ^  j515  A2  B^ 
A3  BV  A,  BU  <fcc.,  A  ^w,  Eq's  (  FA),  be- 
come ;  (since  the  first  and  last  moments  MQ 
and  Jfw  =  0 
A,  ^-S1  =  2Ml  (I,  +  Z 


,  4i  £ 

c%c.,  <fcc.,  <fcc. 

A.,  +  J?,,^  —  JC,  A., + 2  JC,  (4.,  +  4)1 

For  two  continuous  spans,  there  is  but  one 
middle  pier,  and  we  have  this  equation  only  : 
A  2  +  _#!  =.  2  Jfx  (^  +  £2)  and,  in  case  that 
^  —  /2,  finally 

&  (n-m)  (2n-m)  ) 
for  span  II.        f 

,    ^  |  ^l  (w*-«w8) 
^  (  for  span  I. 

The  values  Ml  M2  M^  &c.,  Mn_^  being  found; 
Eq's  (II)  teach  how  to  calculate  the  elastic 
reactions  p  1  p2  p^  &c.,  pn,  which  in  combina- 
tion with  the  static  reactions  of  each  span 
due  to  the  law  of  the  lever  give  the  actual 
reactions  of  the  piers,  that,  may  be  positive  or 
negative,  compression  or  tension. 

The  values  E  and  I  in  Eqs.  (V),  (VI), 
and  (  VII) ,  have  totally  disappeared,  but  the 
suppositions  of  their  being  constant  throughout 


71 


the  whole  bridge  are  embodied  in  the  equa- 
tions, and  without  these  suppositions  being 
fulfilled  the  equations  cease  to  be  correct.  In 
fact,  J?and  Zhave  only  disappeared  because 
we  suppose  them  to  be  constant  values,  and 
should  not  have  disappeared  in  reality. 

Next  we  have  to  make  corrections  of  these 
formulae  for  the  instances  that  the  continuous 
girder  does  not  properly  fit  to  its  bearings,  on 
end  or  middle  piers.  Such  misfits  may  arise 
from  settling  of  the  piers,  from  bad  manufac- 
ture of  the  iron  trusses,  or  from  the  effect  of 
the  rays  of  the  sun  being  greater  on  one  chord 
than  on  the  other,  or  from  winds  cooling  one 
chord  sooner  than  the  other.  This  investiga- 
tion, which  simply  consists  of  a  reapplication 
of  the  principle  of  continuity,  will  give  us 
another  opportunity  to  show  how  simply 
these  problems  can  be  solved  with  our  method. 

Suppose  A  B  to  be  a  straight  line  drawn 
through  two  end  bearings  of  a  continuous 
girder,  and  that  d±  d9  dz  denote  the  depres- 
sions or  elevations  of  the  middle  piers,  posi- 
tive in  case  of  elevation,  negative  in  case  of 
depression.  Further,  suppose  the  continuous 
girder  to  be  cut  into  n  single  spans,  freely 
placed  on  their  supports.  The  problem  then 


72 


is  this,  which  additional  or  correctional  mo- 
ments M,  MV  &G.  Mn  are  necessary  to  again 
connect  the  girders  continuously  ? 

This  problem  at  the 
first  sight  is  nearly  the 
same  as  that  which  we 
have  solved.  In  the 
previous  case,  the  mo- 
ments M1  M2  Ms,  had 
to  lift  up  the  single 
spans  in  such  a  manner 
as  to  make  the  sum  of 
deflections  6m  -)-  ym  -{- 1 
—  fim  +  oim  +  r  In 
this  problem  there  are 
also  angles  d  and  y\ 
but  6  and  y  of  each 
span  are  equal  in  value, 
which  in  the  problem 
just  solved  was  not 
necessarily  the  case. 
Again,  the  angles  d 
and  y  in  the  solved 
problem,  were  below 
the  horizontal  line 
drawn  through  the  middle  pier,  whose 
equation  (I)  was  under  examination.  In  the 


present  problem,  those  angles  may  be  above 
the  horizontal  line,  consequently  there  may 
be  cases  when  we  shall  have  to  consider  them 
as  negative. 

There  may  arise  instances  that  one  or  more 
of  these  moments  will  no  longer  draw  to- 
gether the  top  chords  of  the  trusses,  but  push 
them  apart  ;  in  other  words,  the  moments  M^ 
MI  may  bring  pressure  in  top  chords,  and 
tension  in  the  bottom  chords.  We  then  have 

for  our  problem  dm=ym=±  {  d>*~  d<*-i  )   . 

\  I'm  ' 

the  positive  sign  to  be  taken  if  the  leg  of  the 
angle  is  below  and  the  negative  if  the  leg  is 
above  the  horizontal  line  through  the  middle 
pier  under  consideration.  In  all  other  re- 
spects the  problem  is  the  same  as  the  one  we 
have  just  described;  namely,  this  is  the  gen- 
eral equation: 

6  E  I  (ym  +  ;/vh)  =  M^  lm  +  2  Mm 


For  n  spans,  there  are  (n  —  1)  equations  of 
this  kind,  and  the  moments  M0  and  Mn  are 
equal  to  zero.  The  values  y  are  to  be  sub- 
stituted with  their  proper  signs. 

In  the  special  instance  of  two  equal  spans 


e?,   being   an   elevation   of   the   middle   pier 
above  the  line  A,  C,  we  have 


Fig.  12. 


I* / ^  —  7 _. 


*      *     *  .-  li'  it 

'2  =  Oj  —  o2  •=.-  --.     Both  angles  are 

>  the  line  A   C,  and  consequently  posi- 
tive; we  therefore  have 

6  E I  (      d^\  3  E  T  d 

so  that  this  elastic  end  reaction  becomes 
p=3  E^Id      (FZZZ) 

If  d  had  been  negative,  M would  have  been 
a  moment,  causing  (above  the  middle  pier) 
pressure  in  top  and  tension  in  bottom  chord. 

We  now  proceed  to  our  last  theoretical 
problem,  namely,  to  calculate  the  influence 
of  heat  on  one  chord.  Suppose,  therefore,  a 
properly  manufactured  girder  resting  on  sup- 
ports A  B  C  D,  etc.,  with  the  bottom  chord 
covered  by  floor-planks:  the  top  chord  ex- 
pands by  the  heat  of  the  sun,  the  difference 


75 


of  temperature  between  both  chords  being  t 
degrees  Fahr.  In  this  instance,  the  uniform- 
ly heated  top  chord  will  expand 

150,000 

of  its  length  for  each  degree  Fahr.  If  the 
girder  is  first  considered  to  be  without  weight 
it  must  assume  a  flat  arc,  whose  radius  is 
easily  found.  Two  posts  which  originally  were 

Fig.  13. 


V 

^  150,000     -, 
parallel  have  spread  apart • of  the  pan- 

t 
el  length,  and  consequently  1  :  p  :  : 

:  h  or  p  =  150,000  --.     But  the  radius  p  be- 
t 

ing  found,  it  is  easy  to  also  calculate  the  ele- 
vation of  this  flat  circle  above  each  middle 
pier,  and  this  known  the  problem  is  at  once 
reduced  to  the  previous  one. 

Especially  for  two  equal  continuous  spans 
there  if  an   elevation  of  the  girder  equal  to 

I2 

—  ,  which  is  the  natural  position  of  the  girder 

2p 


considered  without  gravity,  the  bed  plate  be- 

Z2 
ing  placed    -  -  below  the  bottom  chord  ;  we 

2/J  £2  £ 

have  theref  ore  d=     -  ^^      and    Eq's 

7?  T        t 

nmches),Jf=  - 


P~  ~  1,200,000  I  h  ' 
where  the  minus  sign  indicates  —  regarding 
M,  that  the  moment  causes  compression  in 
the  top,  and  tension  in  the  bottom  chord  — 
and  regarding  p,  that  the  end  piers  really  are 
pressed  by  this  elastic  reaction;  in  other 
words,  that  p  increases  the  pressure  on  the 
end  piers  as  caused  by  dead  and  live  loads  on 
the  girder.  Dead  and  live  loads,  however, 
actually  press  down  the  girder  to  the  middle 
pier  either  partly  or  wholly. 

The  moments  of  correction,  M19  J/"2,  Ms, 
<fcc.,  of  a  girder  being  found,  for  unequal  po- 
sitions of  bed-plates  as  well  as  under  consid- 
eration of  heat  in  one  of  the  chords,  these  re- 
sults have  to  be  represented  on  the  diagram 
sheet  of  moments,  shearing  forces  and  strains, 
and  be  added  algebraically  to  the  moments 
and  shearing  forces  due  to  the  dead  and  live 
loads,  when  it  will  be  seen  whether  these  last 


77 


are  sufficient  or  not  to  cause  pressure  always 
on  the  bed-plates. 

Having  now  laid  down  the  mathematical 
principles  of  the  ordinary  theory  of  continu- 
ity, it  remains  a  mere  mechanical  labor  to 
apply  these  principles  and  their  resulting  for- 
mulae to  any  practical  number  of  continuous 
spans,  which  labor  may,  however,  require 
much  patience.  This  theory  is  founded  on 
the  supposition  that  the  angles  of  deflection 
and  elevation  y,  6,  OC>  fi>  are  not  influenced 
by  the  deformations  due  to  the  web  systems, 
which  assumption  was  about  justified  in  the 
calculation  of  homogeneous  plate  girders  such 
as  we  know  to  have  been  first  used  in  Eng- 
land and  France. 

Is  such  simplification  of  theory  also  jus- 
tified in  case  of  continuous  trusses  of  great 
depth  ? 

It  is  impossible  to  investigate  by  direct 
analysis  this  cause  of  error.  For  we  do  not 
know  the  sections  of  the  web  members,  nor 
can  we  consider  them  of  equal  value,  nor 
could  we  estimate  this  unknown  quantity 
even  if  we  would  assume  it  as  a  constant 
value,  as  was  done  with  the  unknown 
chords. 


78 


All  that  we  could  do  would  be :  first  to  pro- 
portion a  continuous  bridge  under  considera- 
tion of  the  chords  only,  thereupon  to  calcu- 
late the  correction  due  to  the  web  and  then 
make  another  calculation  founded  upon  the 
corrected  sections.  In  this  manner  with  a 
great  deal  of  labor  we  could  finally  succeed 
to  proportion  a  continuous  bridge  properly. 
This  labor  indeed  would  be  immense. 

We  now  shall  develop  the  necessary  for- 
mulas towards  consideration  of  the  deflections 
due  to  the  web  system  of  continous  and  other 
trusses. 

Fig.  17. 


Let  A  B  C  be  a  triangle  whose  sides  a  b  c 
have  been  altered  by  very  small  quantities 
Z/a,  A  b9  Ac;  the  problem  is  to  find  the  alter- 
ation A  oC  of  an  angle.  We  have  a*  =  £2  + 
c*  —  2  b  c  .  cos.  OC,  which,  by  inserting  the 
differences,  leads  to  (a  +  A  a)2  =  (b  +  A  b)* 
+  (c  +  A  c}*  —  2  (b  +  A  b)  (c  +  A  c)  cos.  (oC 

•Moc). 

By  developing  this  equation  and  consider- 
ing that  the  squares  of  differences  are  very 
small  quantities  in  comparison  with  their  first 
powers,  we  get: 


79 

a  A  a=b  A  b  -f  c  A  c  —  (b  A  c  +  c  A  b) 
cos.     OC  -f  b  c  sin.     oC  ^  ex  • 

Hence  we  derive  the  value  of  A  oG 

aAa — bAb — cAc-^tyAc+cAb)  cos.   QC 

b  c  .  sin.   oc 

By  applying  this  formula  to  a  rectangular 
triangle  the  formula  is  simplified  into : 


Fig.  18. 


A  a       Ad       a 
A   a  =  -  ----  .-  .  -=- 
b  d          b 


6  — 


a 


A  R_       ±*      Ab     A  d     jl_ 

b~       ~jjT;?      a          b         } 

The  sum  of  A  oc   -f  A  /3  -f  A  JR  =  0,  as  ex- 
pected. 

These  few  formulae  (l)  are  sufficient  to  cal- 
culate the  angles  of  deflection  at  a  joint  of  a 
properly  built  skeleton  bridge.  . 


80 

Fig.  19. 


Fig.  19  represents  part  of  a  quadrangular 
truss,  whose  panels  have  the  length,  c, 
whose  height  is  h,  and  whose  diagonals 
are  of  the  length  d.  The  truss  being  un- 
der transverse  strain,  receives  alterations  of 
the  lengths  of  its  members  and  consequently 
of  its  angles.  The  angle  A  originally  was 
180  degrees,  we  now  have  to  calculate  the 
small  alteration  y\  of  this  angle.  The  alter- 
ation is  the  sum  of  the  alterations  of  the  three 
angles  around  A,  namely: 

A  cn_,  —  A  cln      A  hnl  —  A  hn  ~] 


h 

-  -  A 


he 


In  this  expression 


A  cn 


h 


(2.) 


repre- 


senting the  influence  of  the  chords  c  and 
c1,  is  a  sum,  because,  if  <Vi  is  under  compres- 
sion, cln  will  be  under  tension  and  the  abso- 


81 


lute  values  of  the  alterations  of  these  quanti 
ties  will  add  together.     The  influence  of  the 

posts  — and    of    the    diagonals 

(z/  dn_i  —  A  dn)  —  —  are  actual  differences 
n   .   c 

of  absolute  numbers.  Hence  it  follows  that, 
generally  speaking,  the  influence  of  the 
chords  on  the  value  of  deflection  must  be 
more  important  than  the  influence  of  the  web 
members. 

The  theory  of  the  elastic  line,  such  as  de- 
veloped with  the  integral  calculus,  throws  off 
the  influence  on  y  caused  by  the  posts  and 
diagonals,  whilst  only  the  chords  are  consid- 
ered. Suppose  the  originally  straight  bottom 
chord  of  a  beam  has  deflected  and  the  angles 
of  180°  at  I,  II,  III.  .  .  .  have  altered  by  the 
values  yv  yv  y^  .  .  .  yn.i,  which  alterations 
here  are  negative  values. 

Fig.  20. 


The  question  arises  which  will  be  the  an- 
gles  x   and  y?     The   originally  horizontal 


82 


chord  pieces  cl  ca  cg  .  .  .  will  form  angles  with 
the  line  A  B,  as  follows: 


Here  also  exists  the  equation  : 


sothat—  y  = 
TJie  chord  pieces  cl  c9  .  .  being  equally  long, 
the  sum  of  the  sines  of  the  angles  which  are 
formed  by  c^  ca  .  .  .  cnl,  with  the  horizontal 
line  A  B  must  be  equal  to  zero.  And  since 
the  angles  are  very  small,  their  sines  can  be 
put  equal  to  the  angles  themselves.  Conse- 
quently we  arrive  at  this  equation  : 

0   =   X  +  (X     +    yi}   4-    (X     -f     yi    -f    yj    + 

(*  +  YI  +  r*  +  y  3)  +  •  •  •  +  (*  +  ri  + 

^  +  ....  +  ^^-0     (4.) 

or, 

—  n  x  =i  (n  —  1)  y  1--J-  (n  —  2)  y2  ~}~  (n~ 

3)  73  +  •  •  •  +  ^  n_2  +  /„.,. 
and  likewise  we  have:  —  n  y  =  (n  —  1)  yn.l 
+  (n  —  2)  y^  -f  (n  —  3)  ;/n.3  +  2/2  +  7,. 
In  case  of  the  span  A  B  being  uniformly 
loaded  and  supported  at  both  ends  the  angles 
x  and  y  would  be  equal,  and  since  the  sum 


83 


x  ~h  y  equals  the  negative  sum  of  the  angles 
yl  -j-  y2  -}-.  }Vi  each  one  would  be  half  this 
sum.  By  inserting  the  values  of  equation  (2) 
in  the  expression  for  x  -f-  y  of  a  uniformly 
loaded  truss,  all  A  hn  and  A  dn  will  disappear, 
with  exception  of  the  influence  of  the  end 
posts  and  of  the  end  diagonals  of  each  sys- 
tem. The  chords,  however,  will  remain  in 
the  formula  for  x  and  y  under  any  circum- 
stances. Of  a  uniformly  loaded  beam,  rest- 
ing upon  two  supports,  the  influence  of  the 
web  on  the  angles  x  and  y  is  as  follows : 

—  x  =:  —  y  due  to  web  =  */2 


in  which  expression  y  h0  and  y  hn  are  nega- 
tive values. 

The  total  expression  is  negative,  so  that  the 
influence  of  the  web  increases  the  angles  x 
and  y. 

The  web  has  a  very  considerable  influence 
on  the  angle  of  deflection  of  a  cantiliver 
beam.  The  originally  straight  beam  A  B 
being  fixed  at  B  is  bent  into  the  curve  Bl,  2, 
3,  4  (7,  when  the  angle  x  will  be  equal  to 


84 
1 


i  \-1 — l)Xi~Kw — 2)yQ-f-..2v 
n   ^  \         //i  7/21         /, 

1* 
+  angle  I  B  A\ 

In  this  sum  no  post  and  no  diagonal  of  the 
truss  A  B  will  disappear. 

Fig.  21. 

A  F''g-2| 


If  we  now  suppose  the  special  case  of  in- 
terest that  the  cantilever  A  B  in  A,  respect- 
ively C  is  acted  upon  by  a  single  weight  P, 
we  know  from  the  law  of  the  lever  that  the 
chords  are  strained  the  more,  the  further  they 
are  from  A.  Hence  we  have  to  multiply  P 
with  its  lever  arm,  from  A  to  the  chord  piece  in 
consideration,  and  to  divide  by  the  section  of 
the  member  and  by  the  modulus.  Hence  the 
value  of  compression  or  extension  of  a  chord 

G     ^P  YL    G 

piece  (7n  equals  -^ — r — \  '    '     ,  •  so  that  the 
Section  X  E-  h 

*  Angle  I  B  A  is  the  angle  which  the  curve  makes  with 
A  B  at  B. 


85 

c2  P 

factor  -y^-- T-? ; —  is  common  to  all  exten- 

E  X  &cfion 

sions  and  compressions  provided  the  sections 
are  taken  as  constant,  which,  as  well  known, 
is  one  of  the  principal  hypotheses  of  the  ordi- 
nary theory  of  continuity.  By  inserting  this 
expression  into  formula  (4)  w  being  the  sec- 
tion, we  get: 

*=-,ic^ 

...  9  +  4  +  ij 

2  c*  P      (n  —  l).n.  (2.  —  l) 
~  n.  Ewh*'~  1.2.3 

If  n  becomes  a  very  large  number 

(n  —  1)   .    (n)   (2n  —  1)  n8 

v '- V        —          turns    into   v 

P.  (en)*  P.I* 

the  angle  x  =  fl     ^  ^  ==  —  -^-j    where 

wh*  2, 

— —  =:  the  moment  of  inertia  /,  and  en  —  / 

ifi 

=  the  length  of  the  span.  This  formula  was 
one  on  which  we  based  our  method  of  the 
treatment  of  the  common  theory  of  continu- 
ous girders.  Its  use  involves  the  supposition 
that  the  extensions  of  the  diagonals,  and  the 
compressions  of  the  posts,  are  immaterial  in 
regard  to  the  angles  of  deflection. 


86 


Since  we  know  that  the  angles  x  and  y 
equal  the  sum  of  all  angles  y,  y  can  readily 
be  derived  from  x,  and  there  will  be 

2c2  .  P        n2  _        2  c2  .  P         n*_ 
y  ~  "  E  .  ic  .  h^  "  ~2         ~E  .  w  .  h*  '  IT 

P.  P 


This  was  the  other  of  the  two  equations  III. 

For  the  purpose  of  this  paper  it  will  be 
sufficient  to  explain  the  use  of  these  formulae 
on  two  equal  continuous  railroad  spans,  by 
which  calculations  we  shall  gain  the  opportu- 
nity to  prove  numerically  the  opinions  laid 
down  in  previous  paragraphs. 

V.  —  CALCULATIONS  OF  THE  STRAINS,  SEC- 
TIONS AND  WEIGHTS  OF  Two  200  FEET  RAIL- 
ROAD SPANS,  compared  under  the  same  speci- 
fication with  a  200  feet  single  span.  Exami- 
nation of  the  question  of  economy. 

Specification.  —  To  construct  two  200  feet, 
square  through  spans,  14  feet  between  trusses, 
of  most  economical  height,  with  iron  cross 
bearers,  and  with  iron  stringers  8  feet  apart. 
For  live  load  consider  a  train  equal  to  2,240 
pounds  per  lineal  foot,  headed  by  a  locomo- 
tive concentrating  on  a  cross  bearer  !2/8  tons 
per  lineal  foot  of  a  16  foot  panel.  For  late- 


87 


ral  and  transverse  stiffness  assume  wind  pres- 
sure of  25  pounds  per  square  foot  acting  on 
the  bridge  when  filled  with  passenger  cars. 
Maximum  direct  strain  in  any  point  of  the 
bridge  to  be  10,000  pounds — shearing  strain 
8,000  pounds — per  square  inch,  and  compres- 
sional  sections  of  columns  with  flat  ends  to 

nz 

be  multiplied  by  the  factor  (1  +  )  where 

OjOOU 

n  represents  the  length  of  member  measured 
by  the  least  diameter  of  gyration  of  a  me- 
chanically well-built  post ;  compression  mem- 
bers with  hinged  joints  to  be  treated  corres- 
pondingly, according  to  theory.  The  con- 
nections of  web  diagonals  and  chords  to  cor- 
respond with  the  supposition  of  the  calcula- 
tion. 

Under  this  specification,  we  divide  the  200 
feet  spans  into  12  panels  of  16  feet  8  inches 
long,  and  we  assume  the  dead  weight  per 
lineal  foot  equal  to  1,200  pounds. 

First. — Calculations  of  a  continuous  bridge 
of  two  spans,  200  feet  each. 

In  accordance  with  the  specification  of  one 
truss  the 

panel  live  load  is  — - —  X     ^T=  18,666  Ibs. ; 

2  L~ 


88 


increment, — '— — =  1,555  Ibs. ; 
12 

panel  dead  load,  -  —  =  10,000  Ibs. ; 

10.000 
increment,  — — —  =   833  Ibs. ; 

12 
2 
ocomotive  excess,  —  X  18,666  =  12,444  Ibs. ; 

o 

12444 

increment,  — — —  =  1,036  Ibs. 
12 

In  the  equation  for  moment  over  the  middle 
pier  (Eq.  VII)  the  following  values  are  to  be 
substituted, for  P,  18,666,  10,000  and  12,444; 
for  ny  12;  for  m,  1,  2,  3,  <fcc.,  to  11;  for  d, 

200        100 

—  =  —  and  for  /,  200. 

The  elastic  reactions  p  must  be  subtracted 

from  the  static  reactions  P -.  and  be 

n 

added   to  the  reactions  of   the  middle  pier 

PtTl 

In  the  table,  calculation  of  the  values 

n 

^n  —  m  .  nm  .  .,  - 
p,  P and  P  •  —  is  carried  out  for  a  pan- 
el dead  load,  a  panel  live  load,  and  a  panel 
locomotive  excess  placed  successively  on  the 
joints  1,  2,  3,  <fcc.,  11,  of  one  span.  The  com- 
binations of  these  values  for  both  spans  lead 


89 


to  the  maxima  reactions  over  end  piers  (A), 
and  over  middle  piers  (  T7).*  They  also  form 
all  material  necessary  to  calculate  the  max- 
ima moments,  not  only  of  M  over  the  middle 
pier,  but  at  any  other  vertical  sections  of  the 
girders. 

We  will  next  calculate  the  maxima  mo- 
ments : 

(a)  Moments   due   to   dead  load.     A  = 
40,170; 

M,  =  2  X  —  ?>415  X  20°  =  ~-  2,966,000 
pound  feet.  Any  moment  Mm,  is  found  by 
considering  the  m  panel  loads  acting  at  their 
joints.  There  is  namely,  in  accordance  with 
the  law  of  the  lever: 

Mm  —  40,170  X  me?  — (1+2  +3  +  <fcc., 

+  (m—  1)  ]    d  X  1.0000. 

200 

or,  Mm  —  [40,170  —  (m  —  1)  5,000]  m.  — 

12 

The  value  of  this  formula  can  be  easily 
measured  on  the  diagram  of  forces.  (See 
Plate.) 

(b)  Curve  of  moments  due  to  full  live  loads. 

*  Simply  on  the  law  of  super  position  of  effects,  following  di- 
rectly from  the  law  ut  extensio  sic  vis ;  also  see  Annales  des 
Fonts  et  chaussees,  1874,  paper  on  continuous  girders  by  M. 
Choron. 


90 


Here  is  A  =  74.866,  F  =  130,466,  p  •=. 
27,800  pounds,  and  Ml  =  200  X  27,800  =z 
5,500,060  pound  feet.  Any  moment  M.m  — 

[74,866  —  (m  —  1)  9,333]  m.~ .  (See  Plate. ) 

i  — 

(c)  Curve  of  moments  due  to  live  load  on  one 
span  only.  A  =  88,766,  p  —  13,900,  Ml  = 
2,780,000  pound  feet,  Mm  =  [88,766  —  (m 

-  1)  9,333]  m.  ~.     (See  Plate.)     The  curves 

thus  obtained  enable  us  to  find  the  max- 
imum moment  for  any  panel  of  the  bridge. 
This  is  done  on  the  diagram  by  adding  the 
positive  and  negative  moments  occurring  at 
any  points.  The  moments  M&,  Mb,  M^  can 
also  be  obtained  by  calculating  or  drawing, 
first,  the  curve  of  moments  under  the  consid- 
eration of  single  spans,  and  then  the  (straight) 
line  of  moments  due  to  the  elastic  reaction  p, 
whereupon  the  difference  of  these  values,  for 
any  point  m,  agrees  with  the  values  calcu- 
lated as  above. 

The  maxima  shearing  forces  are  now  to  be 
calculated : 

(a.)  Shearing  forces  due  to  the  dead  load. 
These  forces  can  be  easily  obtained  by  sub- 
tracting 1,  2,  3,  &c.  (n — 1),  panel  dead  loads 


88? 


SSSSSoS 


-iWkor— Oii— i«M-<*>otcr^. 


£ 


-^-i—  < 

CClOOO^CSC^COGOO 


SOffl-^lOt^ 


P 


ocicjxoioa 


O 
H 


fc 

I 


Q 

I 

Q 

-tj 
H 

q 


s  s 

• 


Slilllll 


ilii 


&8§^iS:§  |g 


glilil 


;n5ooS«3Scoocoao 


s^S. 


92 


from  the  value  A  =  40,170.  This  operation 
is  represented  on  the  diagram  of  shearing 
forces. 

Fi£.  14. 


Loaded 

Yimrmet^  
I*-4 

(b.)  Maxima  live  shearing  forces  acting 
in  the  direction  of  A.  For  this  purpose,  the 
second  span  is  supposed  to  be  unloaded,  and 
a  train  to  extend  from  the  panel  m  to  the 
middle  pier. 

Fig.  15. 


Oh**** 

^,-r 

|  1  _= 

<..  /__•  ^ 

M 

(c.)  Maxima  live  shearing  forces  acting  up- 
ward in  the  center.  For  the  calculation  of 
these  forces  the  second  span  is  supposed  to  be 
loaded,  and  the  first  span  also  loaded  from 
the  end  pier  to  panel  m. 

The  bridge  of  which  we  are  calculating  the 


93 


forces  is  shown  with  two  web  systems.  It 
has  been  explained*  why  it  is  impossible  to 
calculate  exactly,  the  strains  occurring  in  each 
one  of  these  systems.  It  must  now  be  added 
that  this  uncertainty  also  necessarily  attends 
the  chord  strains,  whose  determination  is 
especially  difficult  in  those  cord  pieces  which, 
at  each  passage  of  a  train,  have  to  bear  com- 
pression as  well  as  tension.  There  is  no 
method  to  overcome  this  imperfection  of  the 
theory.  In  the  following  calculation  we  have 
separated  the  systems,  and  have  supposed  that 
each  system  would  act  independently. 

For  the  calculation  of  the  forces  Vc  there 
arises  another  difficulty.  The  value  p  name- 
ly, is  inflenced  considerably  by  the  total  load 
on  the  second  span.  Two  methods  are  possi- 
ble, either  to  combine  the  system,  1,  3,  5, 
&c.,  11  of  the  first  span,  with  the  system  1,  3, 
5,  <fcc.,  1 1  of  the  other  span,  or  with  the  system 
2,  4,  6,  &c.,  10  of  this  second  span.  We 

13  900 
have  assumed,  that  instead  of  — '- —  =  6,950 

somewhat  more,  namely,  8,000  pounds  would 
be  the  value  of  p  (due  to  the  second  span) 
for  each  of  the  systems  of  the  partly  loaded 
span,  Fig.  14. 
*  Page  47. 


94 


The  following  tables  exhibit  the  results  of 
the  calculation,  which  is  made  simply  by  add- 
ing the  respective  values  taken  from  the  pre- 
ceding table. 


SystemO--2--4— 6— 8— 10~12:System    0—1—3—5—7—9—11—12. 


?  T  ?  f  f  f   ' 
t  i  t  i  I  i   ! 


8  8 


j  <X 


h-  co  -a  en  co 

ii 


H-      CO      ^      01      bO 


o    os 


H-1  CO  rf*- OS  GO 

&8i§^i 


CO  00  OS  d  CO  h-> 


COC31CD  «*  OS 


-a  en  os  cc  -q  CD 

CO  CO  GO  GO  OS  4^ 


h-OOOOSCOI- 


tOCDOSCOl-'' 

OH-ccoo-a- 

GO  OS  ~3  CO  CO 


e 

H 


K 

IT1 

O 


tocoosoc^  i  ^H 

doco^osi-';       ^  oM 

OOCOGOGOOO'l  iC  i* 

H-o-a^oi.  j  Og 

~~  i  J?  w 

;  c  oa 


95 


The  shearing  maxima  A  and  V  are  now  to 
be  combined,  so  as  to  obtain  the  total  max- 
ima in  each  panel,  such  as  represented  on  the 
diagram  of  forces.  For  the  calculation  of 
the  diagonal  and  post  strains,  the  two  sys- 
tems again  must  be  treated  separately. 

The  diagrams  for  the  chord  and  web  strains 
in  combination  with  the  two  tables  referring 
to  the  web  strains  of  each  separate  system, 
can  now  be  used  to  calculate  in  the  usual 
manner  the  members  of  the  proposed  bridge. 
For  this  purpose  we  choose  a  height  of  25 
feet  (one-eighth  of  the  span)  and  after  con- 
sideration of  the  web  strains  in  the  end  pan- 
els we  arrive  at  the  diagram  of  strains  repre- 
presented.  An  examination  of  these  strains 
will  give  proof  that  the  chord  strains  cannot 
be  properly  determined  without  consideration 
of  the  diagonals,  and  that  consequently  the 
mere  theoretical  comparison  of  curves  of  mo- 
ments and  shearing  forces  may  lead  to  con- 
siderable errors. 

Top  and  bottom  chords  of  continuous  gird- 
ers after  all,  cannot  be  calculated  with  per- 
fect certainty,  even  under  the  objectionable 
suppositions  made.  It  is,  therefore,  advisa- 
ble to  construct  them  to  resist  tension  as  well 


96 


as  pressure.  A  proper  section  for  these 
chords  would  be  two  built  channels  connect- 
ed with  lattice  bars  at  top  and  bottom.  The 
pins  can  be  put  with  mechanical  correctness 
through  the  centre  lines  of  these  channels, 
and  the  re-enforcements  can  be  placed  so  that 
the  pins  bear  against  the  metal  added  to  the 
web  plates.  We  construct  the  diagonals 
of  weldless  eye  bars,  and  the  counter  rods 
with  swivels.  This  arrangement  has  a  scien- 
tific advantage.  Each  web  member  carries 
but  one  kind  of  strain;  whereas,  in  bridges 
with  diagonal  web  members  only,  diagonals, 
at  least  near  the  centre  of  the  span,  have  to 
resist  tension  as  well  as  pressure,  and  there- 
fore must  be  designed  to  sustain  the  sum  of 
both.  Moreover,  vertical  posts  are  more  con- 
venient, with  reference  to  the  construction  of 
the  joints. 

The  built  chord  channels  are  calculated  to 
be  16  inches  deep,  and  the  angle  bars  3x3 
inches,  the  latticing  to  be  double  top  and 
bottom.  The  posts  are  also  designed  on  this 
basis,  with  2  rolled  channels  and  latticing; 
their  bearings  are  made  flat  against  the  bot- 
tom and  top  chords;  the  radii  of  gyration 
have  been  duly  calculated. 


97 


The  strains  upon  compression  members  are 
in  exact  agreement  with  the  formula,  the 

(  H*      1 

factor       1  X   rTTT^      increasing  from    1.12 
v  5,000 ) 

of  chords  to  1.82  of  posts;  the  section  of  the 
lightest  post  is  taken  at  8  square  inches. 

On  this  basis  the  sections  of  chords,  diago- 
nals, posts,  counters,  &c.,  have  been  determin- 
ed in  agreement  with  the  specification.  From 
the  strain  sheet  thus  obtained  (see  Plate)  this 
bill  of  materials  is  calculated : 

Chords,   latticing,   joint    and    reinforcing 

plates 85  912   pounds. 

Posts  with  latticing,  top  and  bottom  bear- 
ings, rivets 35  560        " 

Diagonals  and  swivels 43292 

Pins  and  rollers  with  cages 3  850 

Cross-beams,  hangers   and  washer-plates  16  000 

Stringers, 25  600 

Struts  and  portals 6  000 

Lateral  rods 4  000 

Castings  (end   post  feet   and  heads    bed- 
plates, &c,) 5  000 

Floor  bolts  and  washers 3000 


Total  weight  of  one  iron  200  feet  span 228  214        " 

Iron,  per  lineal  foot 1.141  pounds. 

Timber  and  rails 300       " 

Total  dead  load,  per  foot 1.441       " 

Assumed  weight  per  foot..    1.200       " 

A  too  light  dead  load,  therefore,  was  as- 
sumed ;  but  the  error  amounts  to  less  than 


98 


5  per  cent,  on  the  truss  weights  proper,  say 
about  5,000  pounds  in  the  span.  The  cor- 
rected weight  of  the  iron-work  of  this  con- 
tinuous bridge  would  then  amount  to  1,166 
pounds  per  lineal  foot. 

For  the  sake  of  comparison  under  precisely 
the  same  specification,  for  the  same  form  of 
truss,  for  the  same  details  and  the  same  num- 
ber of  panels,  a  200  foot  single  span  has  been 
calculated.  This  is  the  strain  sheet  with  data 
and  mode  of  computation  ;  * 

Span  200  feet  ;  12  panels,  16  feet  8  inches 
long  ;  diagonals  for  27  feet  depth,  31  x  and 
43  feet  ;  secants,  1.17  and  1.59  ;  tangents, 

0.61    and    1.23  ;     dead  load,   l,200X-°4°  = 

10,000;      live     load,      2,240  X^r=  18,666 

24 

pounds  per  panel  ;  excess  of  locomotive  load 
on  a  joint,  12,  444 

Chords.— 3X0.61X28,666=53,000;  2X1- 23, 
X  28,666  =70,600  ; 

IX  1-23  X28,666  =  35,300;  2KX1-23  X  28 
666  =  87,000  ; 


*  This  example  will  show  the  wide  difference  as  to  the  time 
required  for  the  calculation  of  the  strains  of  a  single  span 
bridge  compared  with  continuous  bridges* 


99 


§ 

£ 
g 


QQ 


r 


From  dead 
"  live 
"•  locor 


88S 

fff 


o'^ 

' 


Maxima 
Diogonal 


From  dead 
'•  live 
•* 


Maxima 

Diagona 


100 

I1/!  X  !-23  X  28,666  X  53,000;  »/,  X  !-23  X 
28,  600  =  17,600. 

Addition. — 53,000,  87,000,  70,600,  53,000 
35,300,  17,600  ;  whence  the  chord  strains, 
53,000,  140,000,  210,600,  263,600,  299,000, 
317,000  pounds. 

The  compression  members  are  designed — 
first,  as  done  with  the  continuous  span  ;  and 
second,  to  consist  of  hollow  segment  columns. 
The  same  sections  are  adopted  for  both  cases, 
but  with  the  hollow  posts  we  gain  all  latticing 
and  still  have  a  greater  factor  of  safety  in 
regard  to  ultimate  strength. 

BILL  OF  MATERIALS. 

Top  chords 49,500  pounds. 

Bottom  chords 28,000 

Pins  and  rollers 4,500 

Posts 34,000 

Diagonals 37,000 

Cross  bearers,  &c 16,000 

Stringers 25,600 

Lateral  struts  and  portals 6,000 

Lateral  rods 4,000 

Castings 5,000 

Floor  bolts 3,000 

Totol  weight  of  iron 212,600 

Weight  per  foot 1,063 

Timber  and  rails ...  300 


Total  weight  per  foot 1,363 

Weight  assumed 1,200 


101 

Hollow  colum  chords 39,000  pounds. 

Bottom  chords 28,000 

Pins  and  rollers 5,000 

Hollow  posts 27,300 

Diagonals 37,000 

Cross  bearers 16,000 

Stringers 25,600 

Lateral  struts  and  portals 6,000 

Lateral  rods 4,000 

Castings 12,400 

Floor  bolts 3,000 

Total  weight  of  iron 203,300 

Weight  per  foot 1,017 

Timber  and  rails . 300 


Total  weight  per  foot 1,317 

Weightassumed 1,200 

The  weight  assumed,  1,200  pounds,  con- 
sequently was  too  light  also  for  a  single  span, 
and  the  truss  weight  shoiddjlje  iricfreased  by  ,4 
per  cent.,  so  that  the  actual* weights  woulcf  lie 
respectively  1383  andfl33^,ppun4^|)^^o<)t; 
These  still  are  respectively  5  S'ahcTlO^  pounds 
less  than  we  obtained  for  the  continuous 
girders. 

Having  now  seen  that  in  the  construction 
of  two  continuous  spans  there  is  no  economy, 
if  compared  with  properly  designed  single 
spans,  it  will  be  well  to  examine  the  weights 
in  detail. 

These  following,  are   the   percentages   of 


102 


weights  as  calculated  for  a  supposed  dead 
load  of  1,200  pounds  per  foot. 


CONTINUOUS 
GIRDERS. 

25  FEET  DEEP. 

SINGLE  SPANS 
27  FEET  DEEP. 

Latticed 

Posts. 

Hollow 
Columns. 

Chords  .... 

37.7 
34.6 

36.4 
33.3 

33. 
31.6 

Webs  

Both  
Balance 

72.3 

27.8 

69.7 
30.3 

64.6 
35.4 

This  comparison  shows  that  the  more  per- 
fect the  .detail  design,  the  smaller  the  per- 
centage of  weight  taken  up  by  the  chords  and 
'.&ej?£/  Tbe  sir-gle  span,  with  hollow  wrought 
iron  segment  columns,  gives  the  best  result. 
The  single  span,  with  latticed  posts,  is  super- 
ior to  the  continuous  girders  with  latticed 
posts  and  chords,  for  the  chords  and  webs  still 
contain  2.6  per  cent,  less  of  the  total  weight 
in  the  first  design  than  in  the  second.  This 
is  not  alone  due  to  the  height,  27  feet,  of  the 
single  span,  for  an  increase  in  height  would 
hardly  reduce  the  chords  of  the  continuous 


103 

girder,  since  5/i2  °f  these  chords  cannot  be  re- 
duced in  section  without  lowering  the  heights 
of  chord  members,  and  therewith  reducing 
the  admissible  chord  pressure. 

The  panel  length  being  taken  at  16  feet 
9  inches,  the  truss  height  could  be  increased 
to  33  feet  without  losing  weight  in  diagonals, 
but  the  posts  would  become  considerably 
heavier. 

The  maximum  height  of  a  truss  is  reached 
if  an  increase  in  height  causes  an  increase 
of  total  weight  ;  that  is,  if  by  an  increase 
of  height  the  web  and  lateral  bracing  in- 
creases more  than  the  chords  decrease. 

The  most  perfect  compressional  members 
permit  the  use  of  the  greatest  depth,  since  the 
weight  of  the  posts  is  a  large  part  of  the  total 
weights.  The  single  span,  with  hollow 
wrought  iron  segment  posts  (Phoenix  col- 
umns) therefore,  has  the  smallest  dead  weight. 
From  the  variability  of  strains  in  their  chords 
and  webs,  continuous  girders  require  contin- 
uous riveted  chords.  Under  this  construction, 
loss  of  material  seems  unavoidable,  because 
these  members  cannot  be  made  without  it, 
in  practically  too  small  sections  at  the  points 
where  the  moments  became  zero.  On  the 


104 


other  hand,  it  will  be  found  advisable  in  con- 
tinuous girders  to  avoid  too  great  a  variety 
of  riveted  members  intermixed  with  eyebars. 
This  construction  has  been  tried  several  times 
in  this  country  with  drawbridges,  but  it  is 
doubtful  whether  any  gain  actually  is  obtain- 
ed by  such  design. 

The  continuous  girders,  such  as  here  de- 
signed, have  one  advantage  over  the  fixed 
span,  because  the  poste  have  been  arranged 
with  two  flat  ends,  whereas  the  single  span 
was  designed  with  posts  of  but  one  flat  bear- 


SINGLE 

SPAN; 

CONTINUOUS 

GIRDER. 

Latticed 

Hollow 

Members. 

Columns. 

Posts,  pounds  

35,560 

34000 

27  300 

Diagonals,  *4  • 

43,292 

37  000 

37  000 

78,852 

71,000 

64,300 

Ratio    

1.23 

1.11 

1.00 

Chords—  Pounds  — 

86,000 

78,000 

74,000* 

Batio  

1.16 

1.05 

1.00 

*  With  castings. 


105 

The  foregoing  comparison  shows  that  the 
web  of  the  best  designed  single  span  is  23  per 
cent,  lighter  than  the  web  of  the  continuous 
girder.  Theoretically  (compare  strain  sheets) 
this  advantage  of  the  single  span  amounted 
to  only  12  per  cent. 

Theoretically  the  chords  compare  thus  : 
continuous  girder  4,403,  to  single  span  5,026, 
or  as  1.00  to  1.14.  In  other  words,  for  the 
same  height  of  trusses,  27  feet,  though  the 
continuous  girder  appears  theoretically  to 
save  14  per  cent,  in  the  chords,  in  reality  it 
causes  a  loss.  While  a  oontinous  girder  of 
three  spans,  proportioned  according  to  theory, 
would  show  a  gain  in  the  chords  of  33  per 
cent.,  in  fact  (see  Laisle  and  Schubler)  execut- 
ed examples  of  acknowledged  excellence  of 
design  gave  only  15  to  20  per  cent,  and  this 
gain  is  only  comparative  since  obtained  under 
sacrifice  of  height,  the  depth  being  one  twelfth 
instead  of  one  eighth  the  average  length  of 
the  three  continuous  spans. 

Having  given  the  practical  figures  and 
weights  for  200  feet  spans,  it  yet  remains  to 
show  whether  there  actually  was  any  theoret- 
ical advantage  in  favor  of  continuous  skeleton 
structures.  To  this  end,  from  the  foregoing 


106 

tables,  the  theoretical  quantities  were  calcul- 
ated, consisting  of  the  products  of  the  length 
of  each  member  into  its  maximum  strain,  re- 
spectively into  the  sum  of  positive,  plus  neg- 
ative maximum  strains  (in  case  a  member  has 
to  carry  tension  as  well  as  pressure).  In  order 
also  to  show  that  three  spans,  of  what  in  books 
is  usually  claimed  as  a  more  economical  ar- 
rangement of  length  of  spans,  do  not  give  any 
greater  advantage  than  two  continuous  spans, 
a  bridge  of  600  feet  total  length  has  been 
calculated.  Its  outer  spans  have  1 1  panels  of 
16  8 '  each,  its  middle  span  has  14  panels  of 
16'  8",  so  that  the  same  panel  length  is  con- 
sidered which  we  assumed  in  the  previous 
examples.  This  example  will  give  evidence 
that  Laisle  end  Schubler  are  correct  in  stating 
that  two  continuous  spans  are  about  as  econ- 
omical as  three,  and  consequently  also  as 
economical  as  an  arrangement  of  more  than 
three  span ;  in  other  words,  that  it  is  sufficient 
to  confine  our  calculations  to  two  continuous 
spans. 

We  get  the  following  theoretical  quantities 
in  pound-feet,  and  hence  by  multiplying  with 

-r-,  and  dividing  by  the  unit  strain  of  10,000 
3 


107 

pounds  per  square  incfe,  also  the  theoretical 
weight  of  the  trusses. 

THEORETICAL  QUANTITIES. 


718,670  I 


Single  Spans 

277  deep  (not  deep  enough). Webs 

Two  continuous  spans Chords  790,500  \ 

25'- deep Webs 

Three  continuous  spans Chords 

25?  deep  (a  little  too  deep).. Webs 


Pound, 
Feet. 


Chords  837,670  \ 


Total. 


Ibs 


1,555,340.519  100 


100.5 


77^830  |  1,563,330  521  : 
88^000!  1.708,000570109.7 


Per 


It  might  be  rejoined  that  the  advantages  of 
continuity  better  present  themselves  in  case 
of  heavier  dead  loads.  Therefore,  under  pre- 
cisely the  same  conditions,  but  for  a  dead 
load  of  2,400  Ibs.,  we  have  calculated 
two  single  spans,  and  three  continuous  spans. 
These  are  the 


(N.  B.— In  the  three  continuous  spans  the  eflect  of  the 
heavy  locomotive  is  not  yet  considered.) 


108 

THEORETICAL  QUANTITIES. 


I- 

P*  CQ 

Pound, 

£<   £ 

Total. 

.^f  <*H 

Feet. 

^_ 

lb° 

Per 

cent 

Single  Spans,  2007,  277  ( 

1,130,000  \ 
915,000  ) 

2,045,000 

682 

101.4 

Locomotive  considered  (  Webs 

Three  continuous  spans  \ 
257  deep                        j  Chords 

958,400  \ 
1,058,200  j 

2,016,609 

672 

100.0 

Locomotive  not  consid-  1 
ered  .                            i^Webs 

In  this  instance  the  continuous  girders  are 
designed  too  deep,  and  the  single  spans  too 
shallow,  for  their  proper  heights  the  quantities 
of  the  chords  should  have  become  nearly- 
equal  to  those  in  the  webs  ;  and  by  also  con- 
sidering the  locomotive  load  the  theoretical 
advantage  of  1.4  per  cent,  would  have  been 
turned  the  other  way. 

In  all  these  examples,  the  chords  of  the 
continuous  girders  can  be  noticed  to  be  light- 
er than  those  of  single  spans,  whilst,  revers- 
edly,  the  webs  of  continuous  girder  are  heavier 
than  those  of  single  spans,  in  such  proportions 


109 

that  the  gain  in  the  chords  is  just  about  neu- 
tralized by  the  loss  in  the  webs. 

It  must  be  expressly  stated  that,  if  it  were 
possible  with  continuous  girders  to  save  so 
much  in  the  chords,  that  this  saving,  less  the 
extra  weight  in  the  webs,  would  leave  a  final 
saving  ;  this  would  only  indicate  a  saving  in 
the  theoretical  value  of  the  trusses.  The  con- 
necting parts,  as  latticing,  rivets,  reinforcing 
plates,  then  the  lateral  strutting,  lateral  dia- 
gonals, and  the  whole  floors  remain  constant 
quantities  unaltered  by  the  principle  of  con- 
tinuity. In  our  example,  about  one-third  of 
the  iron  of  the  whole  bridge  is  a  constant 
quantity,  and  a  theoretical  saving  in  the 
trusses  of  three  per  cent,  could  only  realize 
two  per  cent,  on  the  iron  work  of  the  whole 
bridge.  Most  writers  on  continuity,  however, 
only  mention  the  theoretical  saving  in  the 
chords,  without  substracting  the  loss  in  webs, 
or  without  considering  the  quantity  of  con- 
stant weight  of  iron. 

The  reason  why  the  webs  of  continuous 
bridges  must  become  heavier  than  those  of 
single  spans,  can  easily  be  demonstrated  by 
the  following  examination  of  a  uniformly  and 
fully  loaded  continuous  bridge  : 


110 


Be  A  £  a  span  of  a  continuous  brige,  uni- 
formly and  fully  loaded  by  p  pounds  per 
lineal  foot,  C  D  may  represent  the  line  of 
shearing  forces,  A  C  being  the  reaction  on 

Fiff.  21. 


AC  ~p. 


.  (l-X) 


Ill 

A,  equal  to  px,  and  B  D  being  the  other  re- 
action, •=.  (I  —  x)  p. 

It  is  well  known  that  of  a  continuous  bridge, 
even  if  uniformly  loaded  (for  instance  by 
dead  load),  save  in  the  middle  span  in  case 
of  an  odd  number  of  spans,  the  reactions  A  and 
B  arising  from  the  load  on  A  B  are  not  equal 
(on  account  of  the  couple  of  forces  +  Y  —  Y* 
see  above,  part  I).  Consequently  the  line  of 
shearing  forces  CD  does  not  pass  through  the 
centre  point  of  A  B. 

The  quantity  required  in  the  web  of  span 
A  B  is  equal  to  the  sum  of  the  triangles  A  GE 
+  EBDy  multiplied  with  a  certain  constant 
co-efficient  dependant  on  the  nature  of  the 
design  of  the  web  system. 

Since  then  A  E=x  and  BE^=i  I — xy  there 
will  be  the  web  —  co-efficient  X  [»f +(^ — a*)f]« 

This  is  no  constant   function  ;    it    has    a 

minimum   which  occurs  for  x  =  l — x-=. 

in  other  words  : 

The  theoretical  value  of  the  web  of  a  single 
span,  even  for  uniform  load,  is  lighter  than  it 
would  be  for  a  continuous  bridge. 

For  single  spans  there  is  the  web  ~  co- 

l* 
efficient  X  ~^>  an(i  f°r  two  equal  continuous 

2 


112 

spans  each  one  is  =  co-efficient  X  0.531,  £2, 
this  being  6.2  per  cent,  more  than  for  single 
spans.  Practically,  in  case  of  many  spans  the 
webbing  of  continuous  bridges  would  only  be 
slightly  greater  than  for  single  spans ;  where 
it  not  that  the  movable  load  influences  con- 
tinuous girders  very  materially  more  than  it 
does  single  spans.  This  is  due  to  the  fact 
that  the  point  E  in  the  previous  figure,  on  ac- 
count of  the  variable  and  moving  live  load, 
moves  over  a  considerably  greater  part  and 
further  from  the  center  of  the  span  AB, 
than  happens  for  single  spans  (compare  strain 
sheet,  pages  94  and  99). 

Now,  taken  as  granted  that  the  theoretical 
web  material  of  continuous  bridges  is  greater 
than  that  for  single  spans,  it  follows  directly 
that  single  spans  can  always  be  designed, 
which  require  not  more  total  material  than 
that  of  a  continuous  bridge.  For  it  is  known 
that,  theoretically  speaking,  in  trusses  with 
parallel  chords  the  least  web  material  is  in- 
dependent of  the  height  of  truss.  Consequent- 
ly, even  if  in  both  systems,  the  theoretical 
web  material  would  be  alike,  all  that  would 
be  necessary  would  be  simply  to  make  the 
single  span  trusses  correspondingly  deeper. 


113 

Since,  however,  the  webs  of  continuous  gird- 
ers require  more  material  than  single  spans  ; 
there  is  only  a  want  for  a  slight  increase  of 
height  of  truss  of  single  span  in  order  to  bring 
it  on  the  same  footing  with  the  continuous 
one.  Practically  the  total  minimum  is  about 
reached  if  the  web  material  and  the  chord 
material  are  equally  heavy,  and  this  limit 
happens  therefore  sooner  for  continuous  than 
for  single  span  trusses. 

Hitherto  we  have  developed  the  strains, 
sections  and  weights  of  two  continuous  rail- 
road girders  of  200  feet  length.  We  have 
made  the  incorrect  supposition  that  the  mo- 
ment of  inertia  of  these  girders  is  a  constant 
value.  But  in  reality  the  effective  sections 
of  the  parallel  chords  vary  from  16  to  38 
square  inches.  Therefore,  the  curves  of  mo- 
ments and  the  values  of  shearing  strains 
given  by  our  formulae,  and  on  which  we  based 
the  estimate  of  weight,  are  not  correct.  We 
have  stated  that  the  maximum  moment  over 
the  middle  pier  would  be  15  per  cent,  greater, 
were  the  moment  of  the  inertia  of  the  girders 
so  varied  as  to  produce  an  equal  maximum 
strain  of  10,000  pounds  per  square  inch  of  the 
bridge  totally  loaded,  and  this  difference  in 


114 

continuous  girders  with  three  openings  sinks 
to  seven  per  cent.  In  the  given  example, 
however,  under  the  suppositions  made,  the 
difference  between  actual  and  calculated  max- 
imum moments  is  less  than  15  per  cent. 
What  it  actually  is,  can  be  determined  by  the 
use  of  a  similar  corrected  theory,  but  would 
involve  great  labor,  without  bringing  us 
much  nearer  to  the  actual  strains.  Indeed  the 
chords  being  not  theoretically  varied,  the 
discrepancy  is  small  as  compared  with  other 
shortcomings  of  the  usual  theory. 

This  tedious  correction  was  made  in  some 
instances,  as  for  the  Vistula  bridge,  near 
Dirschau,  and  the  error  was  taken  into  ac-. 
count  in  the  design  of  the  Krementschug 
bridge  over  the  Dnieper  in  Russia.  This 
bridge  was  designed  in  1866,  with  all  possible 
economy.*  It  consists  of  two  parallel  and 
separate  structures,  one  for  roadway  and  one 
for  double  railway  tracks.  There  are  four 

*  By  H.  Sternberg,  Civil  Engineer  and  Professor  in  Karls- 
ruhe. Through  his  kindness  the  author  received  copies  of  the 
design,  strain  sheet  and  estimate,  so  that  the  figures  quoted 
deserve  the  more  confidence,  as  referring  to  the  completed 
worJt  of  not  a  mere  theorist,  but  one  of  much  practical  ex- 
perience in  matters  of  design  and  manufacture.  The  bridge 
was,  however,  built  upon  another  design. 


115 

through  spans,  118  metres  or  3 87  feet  between 
centres,  bridged  by  two  pairs  of  continuous 
girders.  The  calculation  was  based  on  a  dead 
load  of  3,710  and  on  a  live  load  of  3,600  kilos 
per  meter  for  each  track  (respectively  2,480 
and  2,400  pounds  per  foot).  The  weight  of 
the  iron  work  proper,  amounts  to  2,340  pounds 
per  lineal  foot  of  each  track.  There  are  two 
trusses,  10.5  meters  or  34.5  feet  deep,  which 

is  a  depth  —  of  the  span.  The  webs  are  de- 
signed as  stiffened  lattice  work  arranged  in 
ten  systems,  the  diagonals  running  at  angles 
of  45°.  The  diagonals  of  the  meshes  are  2 
meters  or  6.56  feet  long,  equal  to  the  distance 
of  cross  bearers  of  4  feet  2  inches  depth. 
The  rail-stringers  are  of  wood. 

The  following  weights  have  been  calcu- 
lated : 

Chords  of  one  span  of  two  tracks 946,000  pounds. 

Webs     "    "        "       "  "      508,000        " 

Bracing  and  floor  beams,  &c.,      360,000       " 

1,814,000 

Or  2,343  pounds  per  foot  of  each  track. 
These  weights  are  obtained  under  maximum 
strains  of  8,500  pounds  to  the  square  inch 
(600  kilos  per  square  centimeter)  both  for  com- 
pression and  for  tention  of  compressional 


116 

diagonals  and  chords  as  well  as  of  parts  under 
tension,  for  which  latter  ones  the  net  areas 
are  considered. 

The  lightness  of  webs  was  much  furthered 
by  the  diagonal  system  and  by  adopting  the 
same  strain  throughout  as  far  as  practicable. 
This  lightness  of  course  is  secured,  but  only 
under  sacrifice  of  certainty  as  to  the  diagonal 
strains  and  of  the  postulate  of  scientific  de- 
termination of  these  strains.  Nevertheless, 
the  Krementschug  design  is  certainly  one  of 
the  best  proportioned  and  most  economical 
lattice  bridges,  and  gives  evidence  of  its  being 
designed  by  an  engineer  who  knows  how  to 
appreciate  expense  of  manufacture  in  the  mill 
as  well  as  in  the  shops  and  field.  The  bridge 
when  compared  with  other  European  struc- 
tures is  light,  and  comparatively  much  ligther 
than  the  Mainz  bridge  on  Pauli's  plan,  so 
much  the  more  since  the  latter  is  designed  for 
20  per  cent,  lighter  live  load  and  for  33  per 
cent,  greater  strains.  Yet  compared  with 
single  quadrangular  trusses,  designed  with 
the  most  improved  American  details,  the 
Krementschug  bridge  does  not  show  economy 
in  material,  and  in  regard  to  manufacture  and 
erection  it  cannot  compete  with  single  spans. 


117 

A  quadrangular  double  track  truss  •  bridge  of 
proper  proportions  and  details,  400  feet  span, 
can  be  built  with  the  same  weight  of  iron  per 
lineal  foot  for  the  same  live  load,  and  the 
average  maximum  strain. 

In  the  calculation  of  the  Krementschug 
bridge,  the  moments  and  shearing  forces  were 
first  determined  according  to  the  usual  theory, 
whereupon  the  correction  was  introduced  due 
to  the  varied  moment  of  inertia  of  the  struc- 
ture. The  maximum  moment  over  the  central 
pier  under  full  load  of  the  bridge  amounted 
to  12,300,000  kilogrammeters,  but  was  correct- 
ed to  14,  420,000  kilogrammeters.  The  maxi- 
mum moment,  however,  towards  the  middle 
of  each  span  was  only  reduced  by  21/,  per 
cent.  The  chords,  therefore,  by  the  correc- 
rection  were  increased,  as  also  were  the  web 
strains,  slightly* 

We  shall  now  apply  the  formulae  gained 

*  The  design  of  Herr  Sternberg's  bridge  has  this  advantage 
over  many  newly  built  lattice  bridges,  that  the  lengths  of 
compressional  members  are  so  chosen  as  to  permit  their  be- 
ing proportioned  for  crushing  and  not  for  crippling.  The 
chords,  40  inches  wide  and  32  inches  deep,  are  made  only  6 
feet  long  between  panel  joints,  are  braced  laterally  every  6 
feet  in  the  bottom  chords  (cross-bearers),  and  every  12  feet 
on  top,  so  that  the  top  and  bottom  lines  of  each  chord  are 
held  in  position.  The  compressional  diagonals  form  dia- 


118 

for  the  calculation  of  the  influence  of  the 
webs  on  the  reactions  p  of  our  two  continu- 
ous spans  of  200  feet.  We  shall  only  con- 
sider one  part  of  this  labor  by  supposing  the 
bridge  to  be  fully  loaded  while  an  exact  cal- 
culation would  require  to  suppose  all  those 
modes  of  loading  which  we  have  examined 
under  consideration  of  the  chords,  accord- 
ing to  the  common  theory. 

The  chords  of  our  design  are  almost  of  equal 
section.  For  the  sake  of  simplicity  we  there- 
fore apply  the  general  formula  for  the  angles 
s  and  tf  under  supposition  of  an  average 
chord  section.  The  dead  load  is  1200 
pounds  and  the  live  load  is  2240  pounds  per 
foot.  There  results  for  deflection : 

I  '__      _3440X200X200X200X2 

'  O\/O/< 


"2X24X30,000,000X25X25X25  = 
5.87 


2400 


phragms  between  the  vertical  chord  plates,  32  inches  deep, 
for  panels  of  13  and  15  feet,  the  chords  must  be  proportioned 
against  crippling,  their  radii  of  gyration  must  be  correctly 
calculated  and  inserted  in  the  formula.  Thin  vertical  plates 
of  channel  shaped  chords  should  be  secured  in  position  by 
diaphragms,  and  the  lateral  bracing  should  be  properly  cal- 
culated and  proportioned.  The  radius  of  gyration  for  chan- 
nel shaped  chords  of  lattice  bridges  is  very  small,  and  the 
chord  sections  will  increase  considerably,  if  the  specifica- 
tion is  duly  enforced. 


119 

where  the  divisor  2400  is  the  length  of  the 
span  in  inches.  This  angle  of  deflection 

e     r\H 

'- —  must  be  increased  by  the  influence  of 
2400 

the  posts  and  diagonals.     The  result  of  the 

3  79 
calculation  is      '    ~     showing    a    correction 

amounting  to  about  61  per  cent,  of  the  angle 
due  to  the  chords  alone.  The  angle  of  eleva- 
tion caused  by  the  chords  under  action  of  the 
total  force  p  =  42,630  pounds,  such  as  found 
with  the  ordinary  theory,  equals 

42.630  200* 5.82 

3       "    30,000,000.     25s  ~  2400 
2 

This  value  of  elevation  is  so  nearly  equal  to 

5  87 

,  found  to  be  the  angle  of  deflection  due 
2400 

to  the  chords  alone  as  caused  by  the  full  load 
on  the  two  spans,  that — were  it  not  for  the 
web  system — we  should  feel  very  much  satis- 
fied with  the  exactness  of  the  theory. 

But  the  force  p  •=.  42,630,  also  causes  ex- 
tensions and  compressions  of  the  web  mem- 
bers which  result  in  the  angle  of  elevation 


120 

1  7 
— '—   amounting  to  30  per  cent,  of  the  angle 

caused  by  the  chords.     We  have  now  these 
angles : 

Caused  by  chord.  Caused  by  web.      Total. 
5.87  3.79  9.66 


of  deflection 


241)0  2400  2400 

5.82  1.70  7.52 


of  elevation 

The  total  angles  should  be  equal,  but  they 
differ  by  29  per  cent.  The  angle  of  elevation 
is  too  small,  in  other  words  the  force  42,630 
is  by  29  per  cent,  too  small,  or  hence  the  max- 
imum moment  over  the  middle  pier  which  we 
found  to  be  8,130,000  pound  feet  is  by  29  per 
cent,  too  small,  and  the  chord  sections  at  this 
point  should  be  49  inches  instead  of  38  inches. 
The  webs  are  too  strong  at  the  end  piers  and 
too  weak  at  the  central  pier.  The  chords  in 
the  middle  of  the  spans  such  as  designed  are 
too  large.  The  point  of  contrary  flexure  is 
nearer  to  the  center  of  each  span  than  antici- 
pated. The  force  p  being  increased  from 
42,630  to  55,000  pounds.,  the  end  reaction  A 
under  full  load  from  115,036  pounds  de- 
creases to  102,666  pouuds. 

While  according  to   the   common  theory 


121 


0.375  of  the  total  load  should  be  carried  by 
the  end  piers,  the  corrected  theory  only  gives 
0.34 ;  in  other  words,  instead  of  8/8th  only  8/9th 
of  that  load  are  carried  by  the  end  piers  and 
the  maximum  central  moment  for  two  equal 
continuous  spans  under  the  ordinary  theory 

expected  to  be  —  -  p  I*  becomes  only  —  p.  Z*. 

We  should  now  also  calculate  the  influence  of 
the  web  under  other  suppositions  as  to  the 
position  of  live  load.  We  then  should  have 
to  calculate  anew  the  strain  sheet,  we  should 
have  to  correct  the  sections,  and  finally  make 
the  whole  calculation  over  again,  when  again 
the  claims  for  economy  of  continuous  bridges 
were  to  be  examined. 

We  will  not  enter  into  this  labor,  the  inevi- 
table conclusion  being  that  the  common  the- 
ory is  not  sufficient  for  the  calculation  of 
continuous  skeleton  structures.  Its  use  is  con- 
fined to  the  proportions  of  homogeneous  shal- 
low plate  girders.  In  some  instances  contin- 
uous rolled  beams  and  plate  girders  of  uniform 
section  may  be  used  with  advantage  in  build- 
ings and  for  floors  of  bridges  if  vertical  stiffness 
must  be  secured  and  if  the  head-room  is  very 


122 

limited.  But  true  economy  always  points  to 
single  spans. 

The  most  economical  structures  require  but 
very  little  calculation,  so  that  estimates  can 
be  made  within  a  few  hours  without  formu- 
lae or  drawings.  A  practical  engineer  will 
get  along  without  any  formulae.  All  that 
is  necessary  towards  making  a  good  esti- 
mate is  a  piece  of  paper  and  a  pencil  in 
the  hand  of  a  bridge  engineer,  who  in 
the  school  of  practice  has  learned  to  sift  rub- 
bish, both  analytical  and  graphical,  from  the 
few  principles  of  natural  philosophy  which 
are  really  needed,  which  are  commerciably 
applicable  and  from  which,  by  plain  reason- 
ing, special  rules  readily  can  be  derived  when- 
ever desirable. 

It  must  not  be  understood  as  if  analysis 
were  considered  to  be  worthless;  on  the  con- 
trary analysis,  if  not  superficially  applied^  is 
a  most  powerful  thought  aiding  machinery 
and  always  logically  gives  the  correct  answer 
to  a  question,  but  it  does  not  criticise  the  hy- 
potheses, unless  several  contradictory  hypoth- 
eses had  been  underlaid  to  the  calculation. 
The  proper  appreciation  and  limitation  of  the 
power  of  analysis  from  an  engineering  point 


123 

of  view  most  lucidly  has  been  given  by  the  late 
Professor  Rankine  in  the  preface  to  his  ap- 
plied mechanics,  which  we  recommend  to  the 
readers. 

VI. EsiMATE  OP  POSSIBLE  IRREGULARITIES 

IN  THE  STRAINS  OF  THE  TWO  CONTINUOUS  200 
FEET  RAILROAD  SPANS  investigated  in  the 
previous  paragraphs. 

We  will  now  consider  the  irregularities 
caused  in  the  strains  of  continuous  girders  if, 
from  any  reason,  they  do  not  fit  to  their  bed 
plates.  For  this  purpose,  we  refer  to  Fig. 
11  and  to  Eq's  (VIII),  and  assume  first^ 
that  the  masonry  of  the  middle  pier  has  set- 
tled one  inch.  We  have  for  the  moment  of 
correction  and  for  the  correction  of  the  pier 

...      ZEId       ,          M     __ 

reactions  M  = and^?  =  —  ;  E  being 

l>  I 

the  modulus,  assumed  at  30,000,000  pounds, 
I  the  average  moment  of  inertia,  equal  to  say 
7,200  inches  pounds,  arid  I  the  length  of 
the  span  in  feet,  consequently,  M  = 
3.30  000  000.7  200.1 

12.200.200 =1,350,000  pounds  feet. 

This  moment  of  correction  will  produce  pres- 
sure in  the  top  chords  and  tension  in  the  bot- 
tom chords  over  the  middle  pier.  The  re-ac- 


124 
tion  of  each  end  pier  will  be  increased  by 

l~^<&~-  -  6-75°  Pounds.     The  bridge  be- 
2UU 

ing  fully  loaded,  by  reason  of  the  settled 
pier  the  moment  over  this  pier  will  decrease 
from  8,430,000  to  7,080,000,  that  is,  by  16  per 
cent. 

If  the  bridge  were  fully  loaded  only  on 
one  span,  the  maximum  moment  within  this 

span  would  increase  by/)  -—1=  6,750  .  — —  . 

200  =  562,000  pounds  feet.  The  maximum 
moment  of  the  fifth  panel  (see  Plate)  was 
5,952,000  pounds  feet,  and  increases  to  6,514- 
000  pounds  feet.  This  is  an  increase  of  9VS 
per  cent.,  or  about  as  much  as  it  was  expect- 
ed to  save  in  the  chords  under  application  of 
the  theory  of  continuity. 

If  from  any  reason — defective  construction 
in  the  shops,  or  the  middle  pier  being  built 
too  high,  or  the  end  piers  having  settled — 
the  bed  plate  on  the  middle  pier  should  lie 
comparatively  too  high  by  one  inch,  the  cen- 
tral maximum  moment  would  be  increased 
by  1,350,000  pounds  feet,  and  the  total 
strains  over  the  middle  pier  would  be  in- 
creased by  44,000  pounds,  or  by  16  per  cent. 


125 

of  their  calculated  values,  and  the  moments 
within  the  span  would  increase  or  decrease 
correspondingly.  For  every  other  inch  of 
difference  between  bed  plate  and  girder  bear- 
ing the  same  correctional  strains  would  arise, 
for  Eq's  ( VIII)  teach  that  the  corrections 
are  proportional  to  the  values  of  elevation  or 
depression. 

It  follows  then,  conclusively,  that  the  in- 
troduction of  continuous  girders  requires  the 
best  class  of  foundation  and  masonry  for  the 
piers.  Alone  from  this  reason,  practical  en- 
gineers would  not  like  to  use  delicate  super- 
structures like  continuous  girders,  even  if 
these  would  afford  some  economy  of  mat- 
erial, which,  as  we  have  seen,  is  not  the 
case. 

It  was  proposed,  long  ago,  to  improve  con- 
tinuous girders  by  weighing  the  reactions. 
But  the  question  arises,  whether  this  improve- 
ment, which  involves  some  additional  cost, 
is  any  longer  necessary  when  we  know  that 
the  theory  is  of  so  little  practical  value.  It  is 
also  questionable  whether  a  continuous  girder, 
regulated  by  scales  for  one  mode  of  loading, 
would  still  be  properly  adjusted  under  any 
other  position  of  a  moving  train ;  for  we  know 


125 

that  with  each  other  position  of  the  load, 
other  diagonals  will  come  into  action.* 

We  will  next  examine  the  influence  of  the 
sun  on  continuous  bridges  whose  upper  or 
lower  chords  are  covered  by  roadway  planks 
or  otherwise.**  In  this  climate,  the  power  of 
the  sun  is  great,  as  any  one  may  feel  on  a  hot 
summer  afternoon  by  laying  his  hand  on  iron 
exposed  to  the  direct  rays  of  the  sun.  The 
difference  in  heat  of  iron  thus  exposed  or 
shaded  may  be  30°  or  40°  Fahr.  Suppose, 
therefore,  the  bottom  chords  of  our  200  feet 
spans  to  be  covered  by  planks,  what  would 
be  the  correction  needed  for  a  difference  of 
temperature  equal  to  30°  Fahr.  ? 

Under  this  supposition,  the  girders,  consid- 
ered without  weight,  would  rise  so  as  to  form 

25 
part  of  a  circle  whose  radius  is  150,000  . 

80 

=  125,000  feet.     The  rise  in  the  centre  of  a 

*  For  an  ingenious  mode  of  regulating  the  reactions  of  the 
Boyne  bridge,  see  Theory  of  Strains  by  B.  B.  Stooey.  Vol. 
II.,  p.  460. 

**  Single  span  bridges  without  counter  diagonals,  whether 
with  one  or  more  web  systems,  are  entirely  free  from  this 
influence.  Those  with  counters  experience  in  their  region 
extra  strains  of  about  2,000  Ibs.  per  square  inch,  immaterial 
since  the  counters  and  diagonals  in  the 'center  of  single  spao 
bridges  are  made  stronger  than  indicated  by  calculation. 


121 


chord   of   400   feet  woud   be,  consequently, 
' 


2"  oino    ~    '  >  or    '9 

from  Eq's  (  VII  T),  is  known  to  cause  a 
moment  M  —  1,350,000  .  1.92  =  2,592,000 
pounds  feet.  This  has  a  tendency  to 
reduce  the  moment  over  the  middle  pier, 
which,  for  the  unloaded  bridge,  was  found 
equal  to  374,000  pounds  feet,  leaving  still  a 
pressure  in  the  bottom  chords  over  the  mid- 
dle pier  of  393  pounds  per  square  inch  of  sec- 
tion. The  moment  M,  2,592,000  pounds  feet, 
would  cause  additional  pressure  on  the  end 
piers  of  12,960  pounds,  and  additional  strains 
in  end  diagonals  equal  to  about  10  per  cent. 
of  their  maximum  values. 

If  one  span  were  fully  loaded  the  maximum 
moment  of  5,952,000  would  be  increased  by 

13,000.—.  200=il,083,000  pounds  feet,  which 

is  18  per  cent. 

At  the  points  where  the  moments  change 
from  positive  to  negative  comparatively  very 
great  moments  would  be  produced.  Thus,  at 
the  third  panel-  joint  from  the  middle  pier, 
the  greatest  positive  moment  is  2,200,000 
pounds  feet,  which  would  be  increased  by 


128 

/ 

o 

18,000.  — .  200  =  1,950,000   pounds    feet,    so 

that  the  greatest  positive  moment  at  that  point 
would  be  4,150,000  pounds  feet.  In  case  the 
top  chord  were  covered  by  floor  planks,  the 
moment  M^  2,592,000  pounds  feet,  would 
cause  tention  in  the  top  and  compression  in 
the  bottom  chords  over  the  middle  pier.  The 
maximum  moment,  8,430,000  pounds  feet, 
would  be  increased  more  than  30  per  cent.  ; 
so  that  each  degree  Fahr.  would  cause  one 
per  cent,  of  additional  strain.  The  negative 
moment  at  the  second  panel- joint  from  the 
middle  pier  would  be  increased  from  3,2^0,- 
000  to  5,400,000  pounds  feet,  that  is,  by  67 
per  cent. 

If  the  temperature  in  the  top  chords  were 
raised  to  40°  Fahr.  the  central  bearing  could 
no  longer  act  under  the  dead  load  only,  for 
the  truss  would  be  i  inch  above  the  bed-plate. 

Arch  bridges  without  hinges  must  be  de- 
signed under  the  theory  of  continuity,  with 
its  defects  and  unfounded  suppositions. 
Some  of  the  objections  against  this  theory  as 
here  applied  have  peculiar  force — as  the  dif- 
ficulty of  proper  manufacture,  of  close  fit  to 
the  piers,  and  especially  the  influence  of  tern- 


129 

perature.  A  few  exclusively  theoretical 
writers — on  the  authority  of  Oudry's  experi- 
ments, and  of  thermometric  measurements 
at  the  Tarascon  bridge  in  France — deny  the 
influence  of  heat  on  iron  arch  bridges  with 
flat  bearings.  But  they  ignore  the  experience 
gained  with  the  Theis  bridge  in  Hungary, 
which  might  set  at  rest  experiments  with 
the  thermometer.*  This  bridge  changes  its 
bearings  on  the  abutments  daily,  and  it  is  ob- 
served that  the  pressure  moves  from  the 
lower  chord  bearing  to  the  upper  and  back 
again.  The  bridge  being  by  design  very 
stiff,  leaves  its  thrust  bearings  in  winter,  and 
unloaded,  acts  as  a  beam.  It  was  anchored 

*AIr  Stoney  thus  remarks  about  the  effect  of  temperature  : 
The  rise  in  the  crown  of  one  of  the  cast  iron  arches  of  South- 
wark  bridge  for  a  change  of  temperature  of  50°  Fahr.  was 
observed  by  Mr.  Rennie  to  be  about  1.25  inches;  the  length 
of  the  chord  of  the  estrados  is  24G  feet,  and  its  versed  sine, 
23  feet  1  inch,  and  accordingly  the  length  of  the  arch,  which 
is  segmental,  is  3,020.8  inches.  The  range  of  temperature 
to  which  open  work  bridges,  through  which  the  air  has  free 
access  are  subject:  in  this  country,  seldom  exceeds  81W  Fahr. 
The  range  of  temperature  of  cellular  flanges,  may,  however, 
exceed  that  mentioned  above,  as  Mr.  Clark  mentions  that 
the  temperature  of  the  Britannia  tubular  bridge,  before  it 
was  roofed  over,  differed  "  widely  from  that  of  the  atmos- 
phere in  the  interior,  for  the  top  during  hot  sunshine  has 
been  observed  to  reach  120°  Fahr.,  and  even  considerably 
more ;  and  on  the  other  hand,  a  thermometer  on  the  surface 
of  the  snow  on  the  tube  lias  registered  as  low  as  16°  Fahr." 


130 

to  the  abutments  the  next  summer  after  this 
observation  was  made,  but  during  the  follow- 
ing winter  the  piers  commenced  to  move, 
wherepon  the  connections  were  removed. 
This  example  illustrates  one  of  the  practical 
difficulties  inherent  to  continuous  girders. 

Most  all  European  continuous  bridges,  as 
well  as  single  span  bridges,  have  either  their 
bottom  or  their  top  chords  protected  from  the 
sun's  rays.  The  high  iron  viaducts  in  Switz- 
erland and  France  (Freiburg,  Busseau,  Cere, 
<fcc.,)  have  superstructures  of  7,  6  and  5 
continuous  spans,  of  which  the  top  chords 
are  protected  by  flooring.  But  we  have 
seen  that  the  strains  of  continuous  bridges, 
whose  bottom  chords  are  overheated,  will 
be  greatly  disturbed,  and  that  for  a  dif- 
ference in  temperature  of  30°  Fahr.  addition- 
strains  of  30  and  even  more  than  50  per  cent, 
may  arise.  We  are  not  aware  that  this  has 
been  considered  in  the  construction  of  con- 
tinuous bridges. 

Stone  arches  are  affected  in  the  same  way  as  iron  arches. 
With  increased  temperature  the  crown  rises,  and  joints  in 
the  parapets  over  the  crown  open,  while  others  over  the 
springing  close  up.  The  reverse  takes  place  in  cold  weather. 
In  addition  to  the  longitudinal  movements  to  which  all  gir- 
ders are  subject  from  change  of  temperature,  tubular  girders 
move  vertically  or  laterally  whenever  the  top  or  one  side  be- 


131 

comes  hotter  than  the  rest  of  the  tube.  Referring  to  the 
Britannia  tubular  bridge  Mr.  Clark  states  that  "  even  in  the 
dullest  and  most  rainy  weather,  when  the  sun  is  totally  invis- 
ible, the  tube  rises  slightly,  showing  that  heat  as  well  as  light 
is  radiated  through  the  clouds.  In  very  hot  sunny  days  the 
laterial  motion  has  been  as  much  as  3  inches,  and  the  rise 
and  fall  2.3  inches.  These  vertical  and  lateral  motions  have 
not  been  much  observed  in  lattice  or  open  girder  work,  no 
doubt  because  the  air  and  sunshine  have  free  access  to  all 
parts  (?)  and  thus  produce  an  equable  temperature." 

James  Hodges,  in  his  work  on  the  Victoria  bridge,  remarks : 
"  In  building  the  tubes  the  greatest  increase  of  camber  which 
occurred  in  one  day  consequent  upon  the  difference  of  tem- 
perature between  tops  and  bottoms  of  tubes  was  \%  inch; 
the  barometer  on  the  top  reading  124°,  in  shade  at  bottom  90°, 
making  a  difference  of  34°.  The  thermometer  during  the 
previous  night  was  as  low  as  57°.  It  is  therefore  only  fair  to 
infer  that  as  the  bottom  was  in  shade  it  would  not  be  of  the 
same  temperature  as  the  atmosphere,  and  that  the  increase 
of  camber  of  1#  inch,  was  due  to  difference  of  temperature 
of  probably  as  much  as  50°  Fahr. 

The  greatest  longitudinal  movement  of  roller  beds  was,  for 
258  feet  3%  inches,  due  to  a  variation  of  from  —  27°  to  +  128° 
=  155°  Fahr.  The  greatest  lateral  movement  caused  by  tem- 
perature was  Ik  inch. 

The  Victoria  bridge  is  roofed  over  with  timber  and  tin,  and 
the  temperatures  measured  probably  were  only  those  of  the 
atmosphere,  but  not  those  of  the  iron.  The  girders  being 
only  imperfectly  continuous,  calculation  of  deflection  of 
these  girders  is  still  less  reliable  than  of  theoretically  de- 
signed work.  Also  draw-bridges  (continuous  over  two 
spans)  move  sideways  in  consequence  of  one  truss  being 
more  heated  than  the  other.  The  side  movement  of  a  360 
feet  draw  was  noticed  as  much  as  1#  inch  and  caused 
some  trouble  in  locking.  About  equal  vertical  movements 
were  noticed  of  a  360  feet  draw,  with  planked  floor.  Mr.  C. 
Shaler  Smith  (Transactions,  vol.  Ill,  page  131,  Ac.)  noticed 
alterations  of  the  height  of  support  caused  by  the  sun  and 
reversedly  by  cold  winds— of  as  much  as  1  inch.  Mr.  John 


132 

Griffen  informes  us,  that  a  134  feet  draw  on  the  Philadelphia, 
Wilmington  &  Baltimore  R.  R.  was  so  much  aflected  by  the 
unequal  heating  that  it  could  not  be  turned,  the  deflection 
oeing  about  %,  inches.  This  he  remedied  by  covering  the 
top  chord  with  wood. 

We  still  have  to  compare  the  deflections  of 
single  and  continuous  spans.  According  to 
the  theory,  the  deflection  of  a  single  span 
reaches  its  maximum  under  full  load,  which 


5 

then  is  —     ~^=-r  in  which  P  is  the  total  load. 
384  JcL  JL 

For  two  continuous  spans  the  maximum  de- 
flection occurs  if  only  one  span  is  loaded, 

4     PI5 

and   equals  --  J-rr    These    applied   to   the 
384  JtL  JL' 

two  hundred  feet  spans  give  a  deflection 
under  live  load  for  the  continuous  span 
of  2.08  and  for  the  single  span  of  2.05 
inches.  The  deflections  are  equal,  so  that 
also  from  this  consideration  there  is  no  rea- 
son to  prefer  continuous  girders.  It  would 
not  be  desirable  to  adopt  these  girders  on  this 
score  even  if  the  deflection  were  one-half  less 
than  for  single  spans,  for  we  build  single 
spans  with  a  camber,  so  as  to  make  the  floor 
just  level  under  full  proof  load. 

It  is  only  with  rafters  and  purlins  of  roofs 
made  of  shape  iron  of  small  depth  that  the 


133 

consideration  of  continuity  may  lead  to  con- 
structive advantages,  and  in  some  instances 
this  consideration  may  be  valuable  in  the  con- 
struction of  floors  of  bridges,  but  for  bridge 
trusses  of  which  we  can  vary  the  sections  and 
can  adopt  a  siuitable  depth,  the  question  of 
reduction  of  deflection  never  arises,  and  con- 
tinuity on  this  score  need  not  be  resorted  to, 
either  with  girder  or  with  arch  bridges. 

Finally r,  we  find  that  the  large  continuous 
bridges  over  the  Vistula  of  six  spans,  over  the 
Rhine  at  Cologne,  over  the  Dnieper  at  Kre- 
mentschug,  over  the  Danube  at  Pest,  each  of 
four  spans,  from  321  to  418  feet  long,  are  only 
continuous  over  one  support.  Theoretically) 
there  is  an  advantage  as  to  the  average  mo- 
ment of  flexure  of  girders  stretched  over  more 
than  two  spans,  but  also  a  greater  quantity 
of  total  webs  needed,  and  practically  there 
is  a  loss  of  material  at  each  change  from  con- 
cave into  convex  flexure.  For  two  spans,  there 
are  two  such  regions,  for  three  spans  their  are 
four,  and  for  four  spans  there  are  six.  There- 
fore, practically  two  spans  are  about  as  favor- 
able in  regard  to  the  moment  as  three  or  four. 
Any  how,  the  longitudinal  expansion  of  the 
girders  limits  their  length.  For  a  maximum 


134 

change  of  temperature  equal  to  150°,  the 
change  of  length  amounts  to  one-thousandth 
of  the  span,  which  for  four  400  feet  spans  am- 
ounts to  1.6  feet,  so  that  at  one  end  of  such  a 
bridge,  provision  must  be  made  for  a  move- 
ment of  0.8  feet  or  9.6  inches. 

This  consideration  probably  has  limited  the 
application  of  continuity  to  only  two  spans 
of  great  dimensions.  There  are,  however, 
some  bridges  in  Germany  of  seven  and  even 
nine  (small)  continuous  spans,  and  in  France 
and  Switzerland  are  some  large  viaducts  of 
five,  six  and  seven  spans  of  150  feet  average 
length.  Most  all  of  these  (viaduct  of  Frey- 
burg,  Busseau,  C£re,  a  bridge  in  Vienna,  &c.) 
were  built  by  iron  works  in  France,  and  by 
Benkisser's  method,  the  girders  have  been 
rolled  over  the  piers,  a  method  probably  pre- 
ferred on  account  of  facility  of  erection,  the 
works  having  a  full  plant  for  the  purpose.  On 
the  other  hand,  most  builders  with  whom  the 
number  of  their  orders  is  a  consideration  only 
second  to  quality  and  reputation,  erect  con- 
tinuous girders  on  carefully  built  false  works. 

Before  closing  this  examination  it  is  still 
mentioned  that  the  correctness  of  the  theory 
of  continuous  girders,  or  of  the  execution  of 


135 

the  work,  the  moduli  previously  being  experi- 
mented upon,  can  be  examined  by  comparing 
the  actual  deflection  with  the  calculated  ones. 

Namely,  in  case  the  reactions  being  cor- 
rectly calculated,  also  the  strains  must  be  cor- 
rect, and  hence  the  extensions  and  compres- 
sions, and  consequently  also  the  deflections 
must  agree  with  the  calculated  ones. 

The  French  engineers,  who  developed  and 
applied  the  ordinary  theory,  knew  this  very 
well,  and  as  cautious  engineers  and  thorough 
theorists,  did  not  accept  the  theory  before  they 
compared  the  deflections.  They  found  them 
to  agree  with  what  they  considered  the  req- 
uisite closeness  of  approximation.  But  those 
structures  were  shallow  plate  girders  built  on 
carefully  prepared  false  works,  and  hence  the 
suppositions  of  the  theory  were  much  more 
nearly  fulfilled  than  happens  with  deep 
skeleton  trusses. 

The  deflections,  as  it  were,  would  therefore 
be  the  test  stone  of  the  correctness  of  the 
calculation. 

One  of  the  continuous  structures  of  the  Swiss 
North  Eastern  R.  R.  (which  is  considerably 
heavier  than  we  build  equally  strong  and 
equally  long  single  spans,)  is  the  Ergolz 


136 

Bridge,  near  Augst,  consisting  of  four  spans 
of  100,  122,  122  and  100  foot,  IT  2NX  deep. 

The  test-load  consisted  of  locomotives 
2,667  pounds  per  foot.  The  theoretical 
maxima  deflections  (the  moduli  we  learn 
to  have  been  experimented  upon,  but  to 
what  extent  is  not  known)  were  : 

14,  19,  19  and  14  millimeters  K i/   \\\ 

The  actual  maxima  deflections  under  strains 
of  mean  velocity  were  : 

11,  10,  11,  12  millimeters. 

The  calculation  hence  was  out  of  the  way 
by  as  much  as  90  per  cent,  to,  in  the  minimum, 
17  per  cent.  If  all  actual  deflections  had  been 
less  in  the  same  proportion,  the  result  might 
have  been  attributed  to  the  modulus  being  so 
much  larger  than  calculated  upon.  We  again 
assume  the  modulus  to  be  a  constant,  value 
in  the  following  algebraical  deduction  : 

Let  ABC  represent  a  continuous  bridge,  A. 
and  B,  &c.  representing  reactions  under  uni- 
form load  on  one  or  more  spans. 
I  denotes  the  length  of  span  A.  B. 
xl  denotes  an  abscissis  for  which  the  theo- 
retical deflection  is  d,  and  this 


137 


#!  denotes  an  abscis- 

sis  for  which  the 

actual  deflection  is 

A  denotes  the  theor- 

0 

\ 

etical   reaction  at 

^s^ 

) 

s             A. 

Aj  Denotes  the  ac- 
tual reaction  at  A. 

N 

r<3 

p  in  the  load  per  foot 
on  A  B  being  fully 
loaded. 

E    is    the    assumed 

constant    modulus 

QQ  | 

of  elasticity. 

_i 

/                    -r    •          i 

I  is  the  constant  mo- 

ment of  inertia  of 

the  bridge. 

We    have   the    well 

r-» 

known  equation  : 

'           A 
\ 

/         ^ 

E  I  -,  =:A.  p.—    —  (- 

fi  /yt                      O                          t\ 

tv  JU               £                 D 

(Constant  =  C) 

SS-^ 

L  *-___v 

v-    and 

138 


E  I  y=±.  (ar-tot-Z 


oecause  : 


for  x=l,  y=o,  and  C=  -  (  A.£-£ 

We  now  can  calculate  the  deflection  d  for  a 
point  whose  abscissis^o^. 

Theoretically  we  ought  to  get  : 


but  practically  we  do  receive  : 

TT  T  fj  -       1  >   3         72™  \         P  0,  4  ^l 

j!toi.at  —  —  ^xl  --i  xj  —  ^ajj  —  ^ 

Now,  E,  I,  ojj,  ^  and/)  Jein^  constant  quanti- 
ties, Aj  cannot  be  equal  to  A,  because  dl  is  not 
equal  to  d.  They  only  could  become  equal  if 
E  were  different  from  what  it  has  been  sup- 
posed in  the  calculation.  This  supposition 
here  falls  away,  because  the  actual  deflections 
of  the  Augst  Bridge  do  not  correspond  propor- 
tionally with  the  theoretical  ones.  Hence,  by 
subtracting  the  equations,  we  get  : 

E  I  (dr-dj=  ^-^(xS—PxJ 

in  words  : 

The  difference  of  actual  and  theoretical  de- 
flection would  be  proportional  to  the  difference 


139 

of  actual  and  theoretical  reaction,  provided  we 
were  right  in  using  the  theory  in  the  calcula- 
tion of  continuous  trusses.  And  we  find  : 

.6.  E.I. 


The  actual  reaction  is  greater  than  the 
theoretical  reaction  by 

—  ~  —  —    .  6  .  E  I,  or  consequently. 

x^        I  x± 

The  actual  reaction  not  being  equal  to  the 
theoretical  one,  the  actual  strains  are  not  the 
theoretical  ones,  or  the  calculation  of  theory 
does  not  correspond  with  reality. 

Whether  this  proves  incorrectness  of  the 
theory,  or  improper  execution,  is  not  open  to  a 
conclusion  from  the  mere  experimental  results 
without  the  values  E  and  I.  But  it  is  suffi- 
cient to  know  that  remarkable  differences  of 
theory  and  practice  do  exist,  and  may  be  ex- 
pected again. 
VII.  —  RECAPITULATION  AND  CONCLUSIONS. 

1°.  —  The  mere  theoretical  calculation  of 
the  curves  of  moments  and  shearing  forces 
of  girders  or  arches  without  proper  consider- 
ation of  proportions,  details  and  cost  of  man- 
facture,  is  exceedingly  fallacious,  and  this 
fallacy  will  be  the  greater,  if  the  theory  by 


140 

which  the  moments  and  shearing  forces  are 
calculated,  stands  upon  false  premises. 

2° — There  is  theoretically  no  saving  in  con- 
tinuous bridges  ovej  most  economically 
arranged  single  spans.  Whenever  single 
spans  can  not  be  built  economically  there  is 
the  place  for  continuous  bridges  of  which  the 
points  of  reversion  of  curvatures  are  fixed  by 
hinges. 

3°. — The  theory  of  continuity  is  based  on 
the  hypothesis  of  a  constant  modulus  of  elas- 
ticity, which,  as  proved,  does  not  agree  with 
the  nature  of  the  material.  It  has  been 
shown  that  the  modulus  of  wrought  iron 
varies  from  17,000,000  to  over  -40,000,000 
pounds  per  square  inch.* 

4° — Even  if  it  were  assumed  that  the  mo- 
dulus had  a  constant  value,  still  a  correct 
theory  would  require  that  there  be  but  one 
system  of  diagonals  in  the  web  of  a  continuous 
girder.  Under  an  arbitrary  supposition,  the 
strains  in  the  diagonals  and  posts  of  contin- 
uous girders  with  two  or  more  systems  can- 
not be  calculated,  but  only  guessed. 

5°. — The  theory  neglects  the  influence  on 
the  moments  and  shearing  forces  caused  by 
*Page  161. 


141 

the  deflections  due  to  the  extensions  of  the 
web  ties,  and  to  the  compressions  of  the  web 
struts.!  The  theory  also  needs  a  correction 
if  the  chords  are  varied. 

6°. — The  correct  application  of  the  prin- 
ciple of  continuity  involves  an  exceedingly 
tedious  labor,  and,  if  generally  introduced, 
would  greatly  impede  the  business  of  bridge 
construction  in  this  country. 

7°. — In  the  determination  of  the  section  of 
chords  and  webs,  it  must  be  considered  that  a 
member  exposed  to  tension  as  well  as  to  pres- 
sure, must  be  proportioned  to  resist  the  max- 
imum tension  plus  the  maximum  pressure. 

8°. — Continuous  girders  require  very  ac- 
curate workmanship,  both  in  the  shops  and 
in  the  field,  which,  if  exacted  by  the  inspect- 
ing engineer,  will  cause  a  greater  expense 
than  that  for  single  spans.  The  connections 
at  the  points  where  the  strains  change  from 
the  positive  to  the  negative  must  be  made 
with  more  care  than  if  tension  or  only  com- 
pression had  to  be  resisted.  Especially,  in 
case  of  riveting,  the  holes  must  in  the  field  be 


t  We  have  proved  that  this  influence  is  considerable,  and 
upsets  the  common  theory. 


142 

rimmed  to  match  perfectly,  the  rivets  placed 
more  closely  and  driven  thoroughly. 

9°. — The  foundations  and  masonry  of  piers 
on  which  continuous  girders  shall  be  placed, 
must  be  of  excellent  quality.  Single  span 
trusses  may,  without  injury,  be  placed  on 
piers  which  have  settled  several  inches;  but 
this  is  not  the  case  with  continuous  girders. 
Engineers  contemplating  the  use  of  continu- 
ous girdere  should  realize  the  necessity  of 
this  provision,  and  previously  estimate  the 
additional  cost  of  substructure. 

10°. — If  it  is  intended  to  roll  continuous 
girders  over  the  piers,  ordering  and  inspect- 
ing engineers  should  examine  carefully 
whether  the  contractor  has  calculated  the 
extra  strains  arising  from  the  weight  of  the 
projecting  cantilever,  has  properly  reinforced 
the  posts  and  introduced  additional  diagonals 
and  chord  material  at  the  points  of  change  of 
flexure. 

11°. — Continuous  girders  improperly  built 
or  placed  on  their  bed  plates,  have  to  resist 
greater  strains  than  contemplated,  which,  for 
one  inch  difference  in  height  of  location  of 
bed  plate,  on  the  middle  pier  of  a  200  feet 
span,  is  increased  by  16  per  cent. 


143 

12°. — If  the  upper  or  the  lower  chords  of  a 
continuous  bridge  are  protected  from  the  di- 
rect heat  of  the  sun,  the  strains  are  much  dis- 
turbed and  (for  30°  difference  of  tempera- 
ture) may,  over  the  middle  pier  be  increased 
30  per  cent,  and  at  the  points  of  change  of 
flexure  more  than  50  per  cent.,  and  the  struc- 
ture may  even  rise  from  the  middle  piers, 
notwithstanding  its  dead  load, 

13°. — The  proportions  of  depth  of  span  to 
height  depend  essentially  on  the  system  and 
on  the  details  used.  The  lighter  theoretical- 
ly and  practically  the  web  oan  be  made,  the 
greater  the  height  can  be  chosen,  which  is 
only  limited  by  the  practicable  length  of  web 
members  and  by  the  calculation  of  the  strains, 
sections  and  weights  due  to  the  effect  of  wind. 
Practically,  the  best  depth  is  obtained  if  an 
additional  foot  increases  the  weight  of  the 
total  structure.  Continuous  girders,  requir- 
ing more  material  in  their  webs  than  single 
spans,  cannot  be  built  as  high.  European 
bridges  having  been  built  too  shallow  for 
single  spans,  as  far  as  economy  is  concerned, 
were  better  proportioned  when  built  continu- 
ously. This  is  one  of  the  reasons  why,  in 


144 

Europe,  continuous  bridges  proved  to  be  the 
lighter. 

Properly  proportioned  single  spans  on  the 
same  system,  at  least  should  be  no  heavier. 

14°. — We  have  found  by  investigating  the 
example  of  two  200  feet  spans  that  properly 
designed  single  spans  with  American  details, 
are  actually  lighter  than  continuous  girders. 
The  bridge  of  Buda-Pest  and  the  Krement- 
schug  bridge  are  examples  of  large  continu- 
ous structures  of  economical  European  con- 
struction, but  they  do  not  compare  either  in 
cheapness  or  quality  with  single  spans,  well 
proportioned,  having  the  most  scientific 
American  details. 

15°. — Continuous  bridges  deflect  as  much 
as  single  spans  of  correspondingly  greater 
depths.  It  has  now  been  proved  that  the 
theory  of  continuity,  most  interesting  as  it  is 
in  a  scientific  point  of  view,  nevertheless 
forms  only  a  part  of  pseudo-science;  being 
based  on  false  suppositions,  it  is  too  delicate 
in  execution  and  under  use,  and  finally,  be- 
cause it  is  not  economical.  Practical  con- 
structions are  designed  with  a  certain  factor 
of  safety.  The  truer  the  theory,  the  easier  its 
suppositions  can  be  fulfilled,  the  less  its  re- 


145 

suits  are  modified  by  disturbing  influences; 
the  less  it  is  influenced  by  the  unreliable  in- 
cidents of  the  application  of  human  labor :  the 
more  reliable  a  construction  will  be,  and  the 
smaller  the  factor  of  safety  may  be  taken. 

It  has  been  shown,  that  by  theory  we  can- 
not gain  greater  perfection  in  practice,  unless 
we  constantly  are  comparing  the  results  of 
our  deductive  investigations  with  experimen- 
tal facts.*  Results  of  such  experiments  on 
executed  continuous  girders  are  not  known. 
All  that  we  have  are  some  notes  on  deflec- 
tions of  finished  bridges  under  test  loads. 
We  usually  learn  that  the  deflections  were 
much  less  than  expected  or  calculated,  which 
is  communicated  as  a '  proof  of  the  excellency 
of  workmanship,  as  if  workmanship  could  re- 
duce the  extensions  or  compressions  of  iron 
— in  other  words,  could  raise  the  modulus. 
In  this  country,  continuous  girders  have  only 
been  used  for  draw-bridges.  The  calculations 
of  the  strains  of  these  continuous  girders  are 
still  more  complicated,  more  delusive,  and 
more  untrustworthy  than  those  made  for 
fixed  bridges,  principally  on  account  of  the 

*  Compare  what  Mr.  B.  Baker  says,  pages  221,  228  and  313, 
"  Strength  of  Beams,  Columns  and  Arches.    London.    1870. 


146 

compressibility  of  the  turning  apparatus  and 
masonry,  as  well  as  on  account  of  irregulari- 
ities  of  end  supports. 

It  is  not  intended  to  enter  into  these  math- 
ematics. It  only  is  mentioned  that,  proba- 
bly, Prof.  Sternierg  was  the  first  engineer 
who  applied  the  theory  of  continuity  to  the 
various  suppositions  as  regards  dead  load  of 
fixed  anS  swinging  draw,  of  partial  and  full 
live  load,  the  dra^jbeing  screwed  up  at  ends  or 
loose,  &c.  The  pivot  bridge,  by  him  was 
considered  as  a  continuous  girder  over  three 
openings,  two  large  outer  spans  and  a  short 
middle  part.  Herr  Sternberg  applied  the  for- 
mula thus  obtained  to  the  draw  of  Kustrin  in 
Prussia.*  The  central  part  of  the  draw  really 
constituting  a  separate  part  of  a  lattice  beam 
composed  of  chords  and  lattice  diagonals,  this 
calculation  was  justified.  With  large  draws, 

*  His  whole  investigation  was  published  in  the  report  of 
the  Kreutz  Kustrin  R.  R.  of  1857.  In  his  lectures  the  profes- 
sor gave  it  in  all  its  essential  features,  when  he  also  men- 
tioned the  idea  of  weighing  the  reactions  of  continuous  gird- 
trs.  In  this  country  Messrs.  Channte  and  Morrison  in  their 
work  on  the  Kansas  bridge  applied  the  theory  to  continuous 
draw  bridges.  Mr.  Shaler  Smith  in  a  very  lucid  article  in 
the  transactions  of  the  Am.  Society  Co.  E.,  1874,  has  explained 
his  mode  of  building  continuous  draws  without  end  reactions 
when  unloaded,  as  first  introduced  by  him. 


147 

as  built  in  this  country,  the  calculation  based 
on  three  openings  can  be  reduced  to  that  for 
two  spans  so  long  as  the  two  centre  diagonals, 
only  necessary  to  give  stiffness  during  the 
movement  of  the  swing  bridge,  are  provided 
with  elastic  sling  loops,  or  any  other  suitable 
elastic  medium.  The  half  loaded  draw  will 
depress  somewhat  the  drum,  the  wheels  and 
even  the  masonry,  and  the  light  diagonals  be- 
ing incapable  of  taking  up  any  great  amount 
of  strain  without  stretching,  the  other  bearing 
on  the  round  pier  will  remain  in  action.  The 
two  chords  above  the  round  pier  will  sensi- 
bly experience  the  same  amount  of  stress. 
It  is  even  admissible  to  slacken  these  diag- 
onals when  the  draw  is  fixed,  so  that  the 
bridge  may  act  by  a  scientifically  correct  gen- 
eral arrangement  of  modified  continuity,  and 
before  the  bridge  is  to  be  turned,  by  some 
arrangement,  the  diagonals  might  be  brought 
into  action.  By  such  construction  the  greater 
part  of  the  weight  would  not  be  thrown  on 
only  a  few  wheels.  The  great  and  unneces- 
sary complicity  of  the  calculation  of  a  bridge 
resting  on  four  supports  can  be  dispensed 
with,  so  much  the  more  since  the  theory 


148 

of    continuous    trusses    deserves    but    little 
confidence. 

In  the  discussion  on  this  subject  (Transac- 
tiens  American  Society  of  Civil  Engineers, 
1876,  Vol.  V.  pages  227  and  228),  the  author 
has  proved,  mathematically,  the  correctness 
of  this  construction.  It  had  already  been 
put  under  test  in  Mr.  A.  P.  Boilers  draw  span 
of  258  feet  over  the  Hudson  at  Troy,  New 
York;  which,  having  no  center  diagonals 
at  all,  even  swings  around  without  central 
bracing,  simply  relying  upon  the  stiffness  of 
the  riveted  chord.  However,  it  sways  a  little 
more  than  the  designer  wished.  Continuous 
draws  can  be  entirely  avoided  by  building 
draws  consisting  of  two  single  spans  to  be 
united  when  the  water  way  has  to  be 
opened.  This  construction  was  studied  and 
worked  out  in  details  at  the  same  time  by  the 
Keystone  Bridge  Company,  and  by  the  author. 
The  detail  constructions  differ  in  that  point, 
that  the  Keystone  Company  applies  hydrau- 
lic presses  at  the  ends  of  the  single  spans  to 
be  worked  from  the  centre,  so  that  the  ends 
being  raised  the  tensile  central  top  chord  bars, 
with  oblong  pinholes,  are  released  of  their 
tension,  whereas,  the  author  raises  the  ends 


149 

from  the  centre  pier  by  shortening  the  dis- 
tance between  the  central  top  pins,  or  by 
lengthening  the  central  bottom  chord  by 
means  of  inserted  hydraulic  presses.  (For 
sketches  and  details  see  the  discussion  refer- 
red to  above). 

The  investigation  which  is  now  finished 
will  surely  not  impair  confidence  in  the  con- 
struction of  bridges  whose  design  is  exclu- 
sively based  on  the  plain  unmistakable  law  of 
the  lever,  which  can  be  calculated  in  a  short 
time,  be  easily  manufactured  and  erected. 

If  engineers  wish  to  build  continuous  gir- 
ders, they  will  do  better  to  use  continuous 
bridges  with  hinges,  (one  kind  of  such  struc- 
tures was  first  proposed  by  Professor  Ritter, 
hingei  in  alternate  spans  were  first  patented 
by  de  Bergne  in  England  in  1865,  then  rein- 
vented by  Gerber  in  Munich  1866,  and  by 
the  author  in  1867  in  this  country,  each  of 
these  inventions  having  been  made  indepen- 
dently from  the  other,)  when  they  will  escape 
all  uncertainties  caused  by  defects  of  theory, 
by  inequality  of  moduli,  by  several  systems 
of  diagonals,  by  inequality  of  heights  of  sup- 
ports, of  additional  strains  caused  by  heat  of 


150 

sun,  &c*  Practical  men  will  welcome  such 
simplicity,  notwithstanding  it  may  not  satisfy 
a  few  mathematicians,  because  the  problems 
connected  therewith,  to  them,  may  not  seem 
sufficiently  interesting. 


*  Mr.  C.  Shaler  Smith  is  just  about  finishing  a  bridge  of  this 
kind  over  the  Kentucky  River,  consisting  of  3  spans  oi  375 
each.  This  system  was  chosen  on  account  of  facility  ot  er- 
ection, but  not  for  the  sake  of  ecenomy,  the  weights  proving 
to  be  the  same  as  for  single  spans. 


%*  Any  booTc  in  this  Catalogue  sent  free  by  mail  on 
receipt  of  price. 


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SCIENTI  FIC     BOOKS, 

PUBLISHED   BY 

D.  VAN   NOSTRAND, 

23  MURRAY  STREET  AND  27  WARREN   STREET, 
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FRANCIS.  Lowell  Hydraulic  Experiments,  being  a 
selection  from  Experiments  on  Hydraulic  Motors,  on 
the  Flow  of  Water  over  Weirs,  in  Open  Canals  of 
Uniform  Rectangular  Section,  and  through  submerg- 
ed Orifices  and  diverging  Tubes.  Made  at  Lowell, 
Massachusetts.  By  James  B,  Francis,  C.  E.  2(1 
edition,  revised  and  enlarged,  with  many  new  experi- 
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ROEBLING  (J.  A.)  Long  and  Short  Span  Railway 
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Imperial  folio,  cloth 25  oo 

CLARKE  (T.  C.)  Description  of  the  Iron  Railway 
Bridge  over  the  Mississippi  River,  at  Quincy,  Illi- 
nois. Thomas  Curtis  Clarke,  Chief  Engineer. 
Illustrated  with  21  lithographed  plans,  i  vol.  410, 
cloth 7  5^ 

TUNNER  (P.)    A   Treatise  on  Roll-Turning  for  the 
Manufacture  of  Iron.     By   Peter  Tunner.     Trans- 
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I 


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ylvania   Steel   Works,   with  numerous  engravings 

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GILLMORE  (Gen.  Q.  A,)  Practical  Treatise  on  the 
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3 


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HE  WSON  ( Wm.)  Principles  and  Practice  of  Embank 
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KING  (W.  H.)  Lessons  and  Practical  Notes  on  Steam, 
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MINIFIE  (Win.)  Mechanical  Drawing.  A  Text-Book 
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ings and  Machinery ;  an  Introduction  to  Isometrical 
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Shadows.  Illustrated  with  over  200  diagrams  en- 
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Amtrican. 


-  Geometrical  Drawing.     Abridged  from  the  octavo 


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STILLMAN  (Paul.)  Steam  Engine  Indicator,  and  the 
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SWEET  (S.  H.)  Special  Report  on  Coal ;  showing  its 
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WALKER  (W.  H.)  Screw  Propulsion.  Notes  on 
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WARD  (J.  H.)  Steam  for  the  Million.  A  popular 
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WIESBACH  (Julius).  A  Manual  of  Theoretical  Me- 
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edition,  with  an  Introduction  to  the  Calculus,  by  Eck- 
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and  902  wood-cut  illustrations.  8vo,  cloth 10  oo 

5 


D.  VAN  NOSTBAND'S  PUBLICATIONS. 

DIEDRICH.  The  Theory  of  Strains,  a  Compendium 
for  the  calculation  and  construction  of  Bridges,  Roofs, 
and  Cranes,  with  the  application  of  Trigonometrical 
Notes,  containing  the  most  comprehensive  informa- 
tion in  regard  to  the  Resulting  strains  for  a  peiman- 
ent  Load,  as  also  for  a  combined  (Permanent  and 
Rolling)  Load.  In  two  sections,  adadted  to  the  re- 
quirements of  the  present  time.  By  John  D;edrich, 
C.  E.  J  llustrated  by  numerous  plates  and  diagrams. 
Svo,  cloth 5  OQ 

WILLIAMSON  (R.  S.)  On  the  use  of  the  Barometer  on 
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metric Hypsometry.  By  R.  S.  Wiliamson,  Bvt 
Lieut. -Col.  U.  S.  A.,  Major  Corps  of  Engineers. 
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POOK  (S.  M.)  Method  of  Comparing  the  Lines  and 
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ALEXANDER  (J.  H.)  Universal  Dictionary  of 
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rica. By  J.  H.  Alexander.  New  edition,  enlarged, 
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WANKLYN.  A  Practical  Treatise  on  the  Examination 
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PORTER  (C.  T  )  A  Treatise  on  the  Richards  Steam 
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POPE  Modern  Practice  of  the  Electric  Telegraph.  A 
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L.  Pope  Ninth  edition,  revised  and  enlarged,  ami 
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•«  There  is  no  other  work  of  this  kind  in  tha  English  language  that  con- 
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EASSIE  (P.  B.)  Wood  and  its  Uses.  A  Hand-Book 
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BLAKE.  Ceramic  Art.  A  Report  on  Pottery,  Porce- 
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BENET.  Electro-Ballistic  Machines,  and  the  Schultz 
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ENGINEERING  FACTS  AND  FIGURES  An 
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67,  68.  Fully  illustrated,  6  vols.  181110,  cloth,  $2.50 
per  vol.,  each  volume  sold  separately 

HAMILTON1.  Useful  Information  for  Railway  Men. 
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Morocco,  gilt 2  oo 

STUART  (B  )  How  to  Become  n  Successful  Engineer. 
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STUART.  The  Civil  and  Military  Engineers  of  Amer 
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portraits  of  eminent  engineers,  and  illustrated  by 
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CTONEY.  The  Theory  of  Strains  in  Girders  and  simi- 
lar structures,  with  observations  on  the  application  of 
Theory  to  Practice,  and  Tables  of  Strength  and  other 
properties  of  Materials.  By  Bindon  B.  Stoney,  B.  A. 
New  and  revised  edition,  enlarged,  with  numerous 
engravings  on  wood,  by  Oldham.  Royal  8vo,  664 
pages.  Complete  in  one  volume.  8vo,  cloth.  .....  ia  50 

SHREVE.  A  Treatise  on  the  Strength  of  Bridges  and 
Roofs.  Comprising  the  determination  of  Algebraic 
formulas  for  strains  in  Horizontal,  Inclined  or  Rafter, 
Triangular,  Bowstring,  Lenticular  and  other  Trusses, 
from  fixed  and  .noving  loads,  with  practical  applica- 
tions and  examples,  for  the  use  of  Students  and  Engi- 
neers By  Samuel  H.  Shreve,  A.  M.,  Civil  Engineer. 
87  wood-cut  illustrations,  ad  edition.  8vo,  cloth...  500 

MERRILL.  Iron  Truss  Bridges  for  Railroads.  The 
method  of  calculating  strains  in  Trusses,  with  a  care- 
ful comparison  of  the  most  prominent  Trusses,  in 
reference  to  economy  in  combination,  etc.,  etc.  By 
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WHFPPLE.  An  Elementary  and  Practical  Treatise  on 
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inventor  of  the  Whipple  Bridges,  &c  Illustrated 
8vo,  cloth 1 4.  oo 

THE  KANSAS  CITY  BRIDGE.  With  an  account 
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ByO  Chanute,  Chief  Engineer,  and  George  Morri- 
son, Assistant  Engineer.  Illustrated  with  five  litho 
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DUBOIS  (A.  J.)  The  New  Method  of  Graphical  Statics. 
By  A.  J.  Dubois,  C.  E.,  Ph.  D.  With  60  illustra- 
tions. 8vo,  cloth 2  oo 

8 


D.  v.^7  NOSTRAND'S  PUBLICATIONS. 


MAC  CORD.  A  Practical  Treatise  on  the  Slide  Valve 
by  Eccentrics,  examining  by  methods  the  action  of  the 
Eccentric  upon  the  Slide  Valve,  and  explaining  the 
Practical  processes  of  laying  out  the  movements, 
adapting  the  valve  for  its  various  duties  in  the  steam 
engine.  For  the  use  of  Engineers,  Draughtsmen, 
Machinists,  and  Students  of  Valve  Motions  in  gene 
rai.  By  C.  W.  Mac  Cord,  A.  M. ,  Professor  of  Me- 
chanical Drawing,  Stevens'  Institute  of  Technology, 
Hoboken,  N.  J.  Illustrated  by  8  full  page  copper- 
plates. 410.  cloth $3  oo 

K1RKWOOD.  Report  on  the  Filtration  of  River 
Caters,  for  the  supply  of  cities,  as  practised  in 
Europe,  made  to  the  Board  of  Water  Commissioners 
of  the  City  of  St.  Louis.  By  James  P.  Kirkwood. 
Illustrated  by  30  double  plate  engravings.  4to,  cloth,  15  oo 

PLATTNER.  Manual  of  Qualitative  and  Quantitative 
Analysis  with  the  Blow  1'ipe.  From  the  last  German 
edition,  revised  and  enlarged.  By  Prof.  Th.  Richter. 
of  the  Royal  Saxon  Mining  Academy.  Translated 
by  Prof.  H.  B.  Cornwall,  Assistant  in  the  Columbia 
J^chool  of  Mines,  New  York  assisted  by  John  H. 
Caswell.  Illustrated  with  87  wood  cuts,  and  one 
lithographic  plate.  Third  edition,  revised,  560  pages, 
8vo,  cloth 7  50 

PLYMPTON.  The  Blow  Pipe.  A  Guide  to  its  Use 
in  the  Determination  of  Salts  and  Minerals.  Com- 
piled from  various  sources,  by  George  W.  Plympton, 
C  E.  A.  M.,  Professor  of  Physical  Science  in  the 
Polytechnic  Institute,  Brooklyn,  New  York,  umo, 
cloth i  5° 

PYNCHON.  Introduction  to  Chemical  Physics,  design- 
ed for  the  use  of  Academies,  Colleges  and  High 
Schools.  Illustrated  with  numerous  engravings,  and 
containing  copious  experiments  with  directions  for 

e'eparing;  them.       By   Thomas    Ruggles    1  ynchon, 
.  A.,  Professor  of  Chemistry  and  the  Natural  Sci- 
ences. Trinity  College,   Hartford      New  edition,  re- 
vised and  enlarged,  and  illustrated  by  269  illustrations 
on  wood.     Crown,  8vo.  cloth 3  oo 

9 


D.  VAN  NOSTRAND'S  PUBLICATIONS. 

ELIOT  AND  STORER.  A  C9mpendious  Manual  of 
Qualitative  Chemical  Analysis.  By  Charles  W. 
Eliot  and  Frank  H.  Storer.  Revised  with  the  Co- 
operation of  the  authors.  By  William  R.  Nichols, 
Professor  of  Chemistry  in  the  Massachusetts  Insti- 
tute of  Technology  Illustrated,  i2mo,  cloth $i  50 

RAM  M  E  LS B  ERG-  Guide  to  a  course  of  Quantitative 
Chemical  Analysis,  especially  c/  Minerals  and  Fur- 
nace Products.  Illustrated  by  Examples  By  C.  F. 
Ramroalsberg.  Translated  by  J.  Towler,  M.  D. 
8 vo,  cloth 2  35 

DOUGLASS  and  PRESCOTT.  Qualitative  Chemical 
Analysis.  A  Guide  in  the  Practical  Study  of  Chem- 
istry, and  in  the  Work  of  Analysis.  By  S.  H.  Doug- 
lass and  A.  B  JYescott,  of  the  University  of  Michi- 
gan. New  edition.  8vo.  In  press. 

JACOB.  On  the  Designing  and  Construction  of  Storage 
Reservoirs,  with  Tables  and  Wood  Cuts  representing 
.sections,  &c,  iSmo,  boards 50 

WATT'S  Dictionary  of  Chemistry.  New  and  Revised 
edition  complete  in  6  vols  8vo  cloth,  $62.00  Sup- 
plementary volume  sold  separately.  Price,  cloth. . .  9  oo 

RANDALL.  Quartz  Operators  Hand- Book.  By  P.  M. 
Randall.  New  edition,  revised  and  enlarged,  fully 
illustrated.  i2mo,  cloth  200 

SILVERSMITH.  A  Practical  Hand-Book  for  Miners. 
Metallurgists,  and  Assayers.  comprising  the  most  re- 
cent improvements  in  the  disintegration  amalgama- 
tion, smelting,  and  parting  of  the  i  recious  ores,  with 
a  comprehensive  Digest  of  the  Alining  Laws  Greatly 
augmented,  revised  and  corrected.  By  Julius  Silver- 
smith Fourth  edition.  Profusely  illustrated.  i2mo, 
cloth 3  oo 

THE  USEFUL  METALS  AND  THEIR  ALLOYS. 
including  Mining  Ventilation,  Mining  Jurisprudence, 
and  Metallurgic  Chemistry  employed  in  the  conver- 
sion of  Iron,  Copper,  Tin,  Zinc,  Antimony  and  Lead 
ores,  with  their  applications  to  the  Industrial  Arts. 
By  Scoffren,  Truan,  Clay,  Oxland,  Fairbairn,  and 

fithers.     Fifth  edition,  half  calf 3  ?• 

10 


D.  VAN  NOSTRAND  S  PUBLICATIONS. 

JOYNSON.  The  Metals  used  in  construction,  Iron, 
Steel,  Bessemer  Metal,  etc ,  etc.  By  F.  H.  Joynson, 
Illustrated,  izrno,  cloth $o  75 

VON  COTTA.  Treatise  on  Ore  Deposits.  By  Bern- 
hard  Von  (Jotta,  Professor  of  Geology  in  the  Koyal 
School  of  Mines,  Freidberg,  Saxony.  Translated 
from  the  second  German  edition,  'by  Frederick 
Prime,  Jr.,  Mining  Engineer,  and  revised  by  the  au- 
thor, with  numerous  illustrations.  &vo,  cloth 4  oo 

GREENE.  Graphical  Method  for  the  Analysis  of  Bridge 
Trusses,  extended  to  continuous  Girders  and  Draw 
Spans.  By  C.  K.  Greene,  A.  M.,  Prof  of  Civil  Engi- 
neering, University  of  Michigan.  Illustrated  by  3 
folding  plates,  8vo,  cloth a  oo 

BELL.  Chemical  Phenomena  of  Iron  Smelting.  An 
experimental  and  practical  examination  of  the  cir- 
cumstances winch  determine  the  capacity  of  the  Blast 
Furnace,  The  Temperature  of  the  air,  and  the 
proper  condition  of  the  Materials  to  be  operated 
upon.  By  1.  Lowthian  Bell.  8vo,  cloth 600 

ROGERS.  The  Geology  of  Pennsylvania.  A  Govern- 
ment survey,  with  a  general  view  of  the  Geology  of 
the  United  States,  Essays  on  the  Coal  Formation  and 
its  Fossils,  ai  d  a  description  of  the  Coal  Fields  ot 
North  America  and  Great  Britain.  By  Henry  Dar- 
win Rogers,  late  State  Geologist  of  Pennsylvania, 
Splendidly  illustrated  \\ith  Plates  and  Kngravn  gs  in 
the  text.  3  vols  ,  410,  cloth  \\ith  Portfolio  of  Maps.  30  oo 

BURGH.  Modern  Marine  Engineering,  applied  to 
PaddiO  and  Screw  Propulsion.  Consisting  of  36 
colored  plates,  259  Practical  Wood  Cut  Illustrations, 
and  403  paues  01  descriptive  matter,  the  whole  being 
an  exposition  of  the  present  practice  of  James 
Watt  &  Co.,  J  &  G  Rennie,  R.  Napier  &  Sons, 
and  other  celebrated  firms,  by  N.  P.  Burgh,  Engi- 
neer, thick  410,  vol.,  cloth,  $25.00;  half  mor. 30  oo 

CHURCH.    Notes  of  a  Metallurgical  Journey  in  Europe. 

By  J.  A.  Church,  Engineer  of  Mines,  8vo,  cloth.. ...     2  oo 

11 


D.  VAN  NOSTRAND'S  PUBLICATIONS, 

40 

BOCJRNE.  Treatise  on  the  Steam  Engine  in  its  various 
applications  to  Mines,  Mills,  Steam  Navigation, 
Railways,  and  Agriculture,  with  the  theoretical  in- 
vestigations respecting  the  Motive  Power  of  Heat, 
and  the  proper  proportions  of  steam  engines.  Elabo- 
rate tables  of  the  right  dimensions  of  every  part,  and 
Practical  Instructions  for  the  manufacture  and  man 
agement  of  every  species  of  Engine  in  actual  use. 
By  John  Bourne,  being  the  ninth  edition  of  *'  A 
Treatise  on  the  Steam  Engine,"  by  the  "  Artizan 
Club."  Illustrated  by  38  plates  and  546  wood  cuts. 
4to,  cloth $1S  oc 

STUART.  The  Naval  Dry  Docks  of  the  United 
Sjates.  By  Charles  B.  Stuart  late  Engineer-in-Chief 
of  the  U.  S.  Navy.  Illustrated  with  24  engravings 
on  steel.  Fourth  edition,  cloth 600 

ATKINSON.     Practical  Treatises  on  the   Gases   met 

with  in  Coal  Mines.     i8mo,  boards 50 

FOSTER.  Submarine  Blasting  in  Boston  Harbor, 
Massachusetts.  Removal  of  Tower  and  Corwin 
Rocks.  By  J.  G.  Foster,  Lieut -Col.  of  Engineers, 
U.  S.  Army.  Illustrated  with  seven  plates,  4to, 
cloth 3  50 

BARNES  Submarine  Warfare,  offensive  and  defensive, 
including  a  discussion  of  the  offensive  Torpedo  Sys- 
tem, its  effects  upon  Iron  Clad  Ship  Systems  and  in- 
fluence upon  future  naval  wars.  By  Lieut. -Com- 
mander J.  S.  Barnes,  U.  S.  N.,  with  twenty  lithe-- 
graphic  plates  and  many  wood  cuts.  8vo,  cloth.. .  .  5  m 

HOLLEY.  A  Treatise  on  Ordnance  and  Armor,  em- 
bracing descriptions,  discussions,  and  professional 
opinions  concerning  the  materials,  fabrication,  re- 
quirements, capabilities,  and  endurance  of  European 
and  American  Guns,  for  Naval,  Sea  Coast,  and  Iron 
Clad  Warfare,  and  their  Rifling,  Projectiles,  and 
Breech- Loading ;  also,  results  of  experiments  against 
armor,  from  official  records,  with  an  appendix  refer- 
ring to  Gun  Cotton,  Hooped  Guns,  etc.,  etc.  By 
\lexander  L.  Holley,  B.  P.,  948  pages,  493  engrav- 
ings, and  147  Tables  of  Results,  etc.  -  8vo,  half  roan.  10  oo 
12 


D.  TAN  NOSTRAND'S  PUBLICATIONS. 


SIRMS.  A  Treatise  on  the  Principles  and  Practice  of 
Levelling,  showing  its  application  to  purposes  of 
Railway  Engineering  and  the  Construction  of  Roads, 


nples  for  setting 

out  Railway  Curves.  Illustrated  with  three  Litho- 
graphic plates  and  numerous  wood  cuts.  8vo,  cloth.  $2  50 
BURT.  Key  to  the  Solar  Compass,  and  Surveyor's 
Companion ;  comprising  all  the  rules  necessary  for 
use  in  the  field ;  also  description  of  the  Linear  Sur- 
reys and  Public  Land  System  of  the  United  States, 
Notes  on  the  Barometer,  suggestions  for  an  outfit  for 
a  survey  of  four  months,  etc.  By  W.  A.  Burt,  U.  S. 
Deputy  Surveyor.  Second  edition.  Pocket  book 
form,  tuck a  50 

THE  PLANE  TABLE.  Its  uses  in  Topographical 
Surveying,  from  the  Papers  of  the  U.  S.  Coast  Sur- 
vey. Illustrated,  8vo,  cloth 2*0 

"  This  worK  gives  a  description  of  the  Plane  Table,  employed  at  the 
U.  S.  Ceatt  Survey  office,  and  the  inauner  of  using  it." 

JEFFER'S.  Nautical  Surveying.  By  W.  N.  Jeffers, 
Captain  U.  S.  Navy.  Illustrated  with  9  copperplates 
and  3 1  wood  cut  illustrations.  8vo ,  cloth. 5  oo 

CH  AU  V  EN  ET.  New  method  of  correcting  Lunar  Dis- 
tances, and  improved  method  of  Finding  the  error 
and  rate  of  a  chronometer,  by  equal  altitudes.  By 
W.  Chauvenet,  LL  D.  8vo,  cloth 200 

BRUNNOW.  Spherical  Astronomy.  By  F.  Brunnow, 
f'h.  Dr.  Translated  by  the  author  from  the  second 
German  edition.  8vo,  cloth 650 

PEIRCE.  System  of  Analytic  Mechanics.  By  Ben- 
jamin Peirce.  4to,  cloth 1000 

COFFIN.     Navigation  and  Nautical  Astronomy.     Pre- 

fared  for  the  use  of  the  U.  S.  Naval  Academy.     By 
rof.  J.  H.  C.  Coffin.  Fifth  edition.  52  wood  cut  illus- 
trations.    i2mo,  cloth 3  50 

NOBLE  (W.  H.)  Useful  Tables.  Compiled  by  W.  H. 
Noble,  M.  A.,  Captain  Royal  Artillery.  Pocket 

form,  cloth 50 

13 


D.  VAN  NOSTRAND'S  PUBLICATIONS. 


CLARK.  Theoretical  Navigation  and  Nautical  Astron- 
omy. Hy  Lieut.  Lewis  Clark,  U.  S.  N.  Illustrated 
with  41  wood  cuts.  8vo,  cloth $3  < 


HASKINS.  The  Galvanometer  and  its  Uses.  A  Man- 
ual for  Electricians  and  Students.  By  C.  H.  Has- 
kins.  i  zmo,  pocket  form,  morocco 200 

MORRIS  (E.)  Easy  Rules  for  the  Measurement  of 
Earthworks,  by  Means  of  the  Prismoidal  Formula. 
By  Ellwood  Morris,  C.  E.  78  illustrations.  8vo, 
cloth i  50 

BECKWI'lH.  Observations  on  the  Materials  and 
Manufacture  of  Terra-Cotta,  Stone  Ware,  Fire  Hrick, 
Porcelain  and  Encaustic  Tiles,  with  remarks  on  the 

Eroducts  exhibited  at  the  London  International  Exhi- 
ition,    1871.      By  Arthur  Beckwith,    C.    E.      8vo, 
paper 60 

MORFIT.  A  Practical  Treatise  on  Pure  Fertilizers,  and 
the  chemical  conversion  of  Rock  Guano,  Marlstones, 
Coprolites  and  the  Crude  Phosphates  of  Lime  and 
Alumina  generally,  into  various  valuable  products. 
By  Campbell  Morn*,  M.D.,  with  28  illustrative  plates, 
8vo,  cloth 2000 

BARNARD.  Ine  Metric  System  of  Weights  and 
Measures.  An  address  delivered  before  the  convoca- 
tion ot  the  University  of  the  State  of  New  York,  at 
Albany,  August,  1871.  By  F.  A  P.  Barnard,  LL  D., 
President  of  Cohimbia  College,  New  York.  Second 
edition  irom  the  revised  edition,  printed  for  the  Trus- 
tees of  Columbia  College.  Tinted  paper,  8vo,  cloth  $  00 

-•  Report  on  Machinery  and  Processes  on  the  In- 
dustrial Arts  and  Apparatus  of  the  Exact  Sciences. 
By  F.  A.  P.  Barnard,  LL.D.  Paris  Universal  Ex- 
position, 1867.  Illustrated,  8vo,  cloth go 

A.LLAN.  Theory  of  Arches.  By  Prof.  W.  Allan,  for- 
merly of  Washington  &  Lee  University,  1 8mo,  b'rds  50 

ALLAN  (Prof  W.)  Strength  of  Beams  undir  Trans- 
verse I  oacls.  By  Piot.  W.  Allan,  author  of  "  Theory 
of  Arches."  With  illustrations.  i8mo,  boards 50 

14 


r>.  VAN  NOSTRAND'S  PUBLICATIONS. 


MYER.  Manual  of  Signals,  for  the  use  of  Signal  officers 
in  the  Field,  and  for  Military  and  Naval  Students, 
Military  Schools,  etc  A  new  edition  enlarged  and 
illustrated  By  Brig  General  Albert  J.  Myer,  Chief 
Signal  Officer  of  the  army,  Colonel  of  the  Signal 
^'orps  during  the  War  of  the  Rebellion,  izmo,  48 
plates,  full  Roan $5  oo 

WILLIAMSON.  Practical  Tables  in  Meteorology  and 
Hypsometry,  in  connection  with  the  use  of  the  Bar- 
ometer. By  Col.  R.  S.  Williamson,  U.  S-  A.  410, 
cloth 2  50 

CLEVENGER.  A  Treatise  on  the  Method  of  Govern- 
ment Surveying,  as  prescribed  by  the  U.  S.  Congress 
and  Commissioner  of  the  General  Land  Office,  with 
complete  Mathematical,  Astronomical  and  Practical 
Instructions  for  the  Use  of  the  United  States  Sur- 
veyors in  the  Field.  By  S.  R.  Clevenger,  Pocket 
Book  Form,  Morocco 2  50 

PICKERT  AND  METCALF.  The  Art  of  Graining. 
How  Acquired  and  How  Produced,  with  description 
of  colors,  and  their  application.  By  Charles  J'ickert 
and  Abraham  Metcal£  Beautifully  illustrated  with 
42  tinted  plates  of  the  various  woods  used  in  interior 
finishing.  Tinted  paper,  410,  cloth 10  oo 

HUNT.  Designs  for  the  Gateways  of  the  Southern  En- 
trances to  the  Central  Park.  By  Richard  M.  Hunt. 
With  a  description  of  the  designs.  410.  cloth 5  oo 

LAZELLE.  One  Law  in  Nature.  By  Capt.  H.  M. 
Lazelie,  U.  S.  A.  A  new  Corpuscular  Theory,  com- 
prehending Unity  of  Force,  Identity  of  Matter,  and 
its  Multiple  Atom  Constitution,  applied  to  the  Physi- 
cal Affections  or  Modes  of  Energy,  izmo,  cloth. . .  r  50 

CORF  T  ELD.  Water  and  Water  Supply.  By  W.  II 
Corfield,  M.  A.  M,  D.,  Professor  of  Hygiene  and 
Public  Health  at  University  College,  London.  i8mo, 
boards 50 

15 


D.  VAK  NOSTKAND'S  PUBLICATIONS. 


BOYNTON.  History  of  West  Point,  its  Military  Im- 
portance during  the  American  Revolution,  and  the 
Origin  and  History  of  the  U-  S.  Military  Academy. 
By  Bvt.  Major  C.  E.  Boynton,  A.M.,  Adjutant  of  the 
Military  Academy.  Second  edition,  416  pp.  8yo, 
printed  on  tinted  paper,  beautifully  illustrated  with 
36  maps  and  fine  engravings,  chiefly  from  photo- 
graphs taken  on  the  spot  by  the  author.  Extra 
cloth  ...........................................  $3  5° 

WOOD.  West  Point  Scrap  Book,  being  a  collection  of 
Legends,  Stories,  Songs,  etc.,  of  the  U  S.  Military 
Academy.  By  Lieut  O.  E.  Wood,  U-  S.  A.  Illus- 
trated by  69  engravings  and  a  copperplate  map. 
Beautifully  printed  on  tinted  paper.  8vo,  cloth  .....  5  oc 

WEST  POINT  LIFE.  A  Poem  read  before  the  Dia- 
lectic Society  of  the  United  States  Military  Academy. 
Illustrated  with  Pen-and-ink  Sketches.  By  a  Cadet. 
To  which  is  added  the  song,  "  Benny  Havens,  oh  1" 
oblong  8vo,  21  full  page  illustrations,  cloth  ..........  a  50 

GUIDE  TO  WEST  POINT  and  the  U.  S.  Military 
Academy,  with  maps  and  engravings,  i8mo,  blue 
cloth,  flexible  ................................  ..  ico 

HENRY.  Military  Record  of  Civilian  Appointments  in 
the  United  States  Army  By  Guy  V.  Henry,  Brevet 
Colonel  and  Captain  First  United  States  Artillery, 
Late  Colonel  and  Brevet  Brigadier  General,  United 
States  Volunteers.  Vol.  i  now  ready.  Vol.  2  in 
press.  8vo,  per  volume,  cloth  ....................  5  oo 

HAMERSLY.  Records  of  Living  Officers  of  the  U. 
S.  Navy  and  Marine  Corps.  Compiled  from  official 
sources.  By  Lewis  B.  Hamersly,  late  Lieutenant 
U-  S.  Marine  Corps.  Revised  edition,  8vo,  cloth...  5  oo 

MOORE.  Portrait  Gallery  of  the  War.  Civil,  Military 
and  Naval.  A  Biographical  record,  edited  by  Frank 
Moore.  60  fine  portraits  on  steel.  Royal  8vo, 
cloth  .............................  •  ..............  6  oo 

16 


D   VAN  NOSTRAND  S  PUBLICATIONS. 

PRESCOTT.  Outlines  of  Proximate  Organic  Analysis, 
for  the  Identification,  Reparation,  and  Quantitative 
Determination  of  the  more  commonly  occurring  Or. 
ganic  Compounds.  By  Albert  B.  Prescott,  Professor 
of  Chemistry,  University  of  Michigan,  izmo,  cloth...  x  75 

PRESCOTT.  Chemical  Examination  of  Alcoholic  Li- 
quors A  Manual  of  the  Constituents  of  the  Distilled 
Spirits  and  Fermented  Liquors  of  Commerce,  and 
their  Qualitative  and  Quantitative  Determinations. 
By  Albert  B.  Prescott;  12 mo,  cloth 150 

NAQUET.  Legal  Chemistry.  A  Guide  to  the  De- 
tection of  Poisons,  Falsification  of  Writings,  Adul- 
teration of  Alimentary  and  Pharmaceutical  Substan- 
ces ;  Analysis  of  Ashes,  and  examination  of  Hair, 
Coins,  Arms  and  Stains,  as  applied  to  Chemical  Ju- 
risprudence, for  the  Use  of  Chemists,  Physicians, 
Lawyers,  Pnarmacists  and  Experts  Translated  with 
additions,  including  a  list  of  books  and  Memoirs  on 
Texicology,  etc.  from  the  French  of  A.  Naquet.  By 
J.  P.  Battershall,  Ph.  D.  with  a  preface  by  C.  F. 
Chandler,  Ph.  D.,  M.  D,.  L.  L.  D.  i2mo,  cloth. ...  2  oo 

McCULLOCH.  Elementary  Treatise  on  the  Mechan- 
ical Theory  of  Heat,  and  its  application  to  Air  and 
Steam  Engines.  By  R.  S.  McCulloch,  8vo,  cloth....  3  50 

AXON.  The  Mechanics  Friend;  a  Collection  of  Re- 
ceipts and  Practical  Suggestions  Relating  to  Aqua- 
ria— Bronzing — Cements — Drawing — Dyes —  Klectri- 
city — Gilding — Glass  Working — Glues — Horology — 


—  Varnishes  —  Water-Proofing    and  "Miscellaneous 
Tools, — Instruments,    Machines   and  Processes  con- 
nected with  the  Chemical  and  Mechanics  Arts;  with         . 
umerous  diagrams  and  wood  cuts.      Edited  by  Wil- 
,atn  E.  A.  Axon.     Fancy  cloth i  50 

17 


D.  VAN  NOSTRAND  S  PUBLICATIONS. 

ERNST.  Manual  of  Practical  Military  Engineering,  Prt 
pared  for  the  use  of  the  Cadets  of  the  U.  S.  Military 
Academy,  and  for  Engineer  Troops.  By  Capt.  O.  H. 
Ernst,  Corps  of  Engineers,  Instructor  in  Practical 
Military  Engineering,  U.  S.  Military  Academy.  192 
wood  cuts  and  3  lithographed  plates.  i2mo,  cloth..  500 

BUTLER.     Projectiles  and  Rifled  Cannon.     A  Critical  - 
Discussion  of  the   Principal  Systems  of  Rifling  and 
Projectiles,  with  Practical    Suggestions  for  their  Im- 

S-ovement,  as  embraced  in  a  Keport  to  the  Chief  of 
rdnance,  U.  S.  A.     By  Capt.  John  S.  Butler,  Ord- 
nance Corps,  U.  ti.  A.     36  plates,  4to,  cloth 7  50 

BLAKE.  Report  upon  the  Precious  Metals:  Being- Sta- 
tistical Notices  of  the  principal  Gold  and  Silver  pro- 
ducing regions  of  the  World,  Represented  at  the 
Paiis  Universal  Exposition.  By  \Villiam  1J.  Blake, 
Commissioner  from  the  State  of  California.  8vo,  cloth  2  oc 

TONER.  Dictionary  of  Elevations  and  Climatic  Regis- 
ter of  the  United  States.  Containing  in  addition  to 
Elevations,  the  Latitude,  Mean,  Annual  Temperature, 
and  the  total  Annual  Kainfall  of  many  locaiiiie?;  with 
a  brief  introduction  on  the  Orographic  and  i  hysical 
Peculiarities  of  North  America.  By  J.  M.  Toner, 
M.  D.  8vo,  cloth 3  75 

MOWBRAY.     Tri-Nitro    Glycerine,   as  applied  jn  the 

do 

observation  and  pn „ 

five  hundred  thousand  pounds  of  this  explo  ive  Mica, 
Blasting  Powder,  Dynamites;  with  an  acc<  unt  of  the 
various  Systems  of  Blasting  by  Electricity,  Priming 
Compounds,  Explosives,  etc.,  etc.  By  George  M. 
Mowbray,  Operative  Chemist,  with  thirtee  n  illustra- 
tions, tables  and  appendix.  Third  Edition.  Re- 
written. 8vo.  cloth 3  01 

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